Skip to main content

Full text of "UICSM newsletter"

See other formats


LIBRARY OF THE 

UNIVERSITY OF ILLINOIS 

AT URBANA-CHAMPAIGN 



5\0.7 
r\o. l"^ 




Digitized by the Internet Archive 

in 2011 with funding from 

University of Illinois Urbana-Champaign 



http://www.archive.org/details/uicsmnewsletter16univ 



v_y i 





e ws et u r 



An occasional publication of the 
UNIVERSITY OF ILLINOIS COMMITTEE ON SCHOOL MATHEMATICS 

1208 W. Springfield 
Urbana, Illinois 



Why a Newsletter? 2 

The Principal Operator 3 
Pattern- sentences, Instances, 

and Consequences 4 

UICSM-NETRC Tests A-D 12 

News and Notices 20 



UICSM Newsletter No. 1 



October 12, I960 



• v.. ^'-•* ■ "■ ■' 



\ 



•••o;j 



..'.'tr 



«*vr,s*r t--!^t. 3(1 



( .v,'/-. 



'■^iv -up M »^«ei '.^ .' 



Why a newsletter? 



The staff of the University of Illinois Committee on School 
Mathematics Project feels there is a need for an improved way 
in which we can communicate with teachers who are using UICSM 
text materials. We already correspond extensively with the 
teachers in participating schools who have been to our training 
conferences, and who send us regular reports on their work, along 
with queries about content and methods of teaching. However, we 
have no way of reaching the many people in nonparticipating schools 
who teach fronn UICSM materials. We therefore decided a news- 
letter such as this might be helpful to many people, and that we 
should begin by sending it to everyone believed by us to be currently 
teaching from First Course materials. 

We have begun by thinking of it as a kind of supplement to, and 
amplification of, the Teacher's Commentary for our various units. 
The major criterion to be met by included materials should be their 
usefulness to the classroom teacher, so we expect to present exposi- 
tory articles relevant to the topics about to be taught in most classes, 
test items, pedagogical suggestions, additional commentary on those 
spots in the text apparently most troublesome to students, and notes 
on the professional activities of UICSM teachers and staff. Perhaps 
later we can also include anecdotes about notable work by students, 
letters from teachers, selected references to non-UICSM publications, 
and maybe even some apt quotations, jokes, and cartoons. 

I have asked Alice Hart and Ron Szoke to take the responsibility 
for the newsletter. Please address contributions and inquiries to 
either of them in care of the UICSM Newsletter, 1208 W. Springfield, 
Urbana, Illinois, --Max Beberman, Director, UICSM Project. 



PRINCIPAL OPERATOR 

Few persons begin difficult undertakings without making arreuige- 
ments to have the proper tool at hand. In addition to having the tool, 
one also wishes to have practice in using it in a less complicated situa- 
tion. In the work on unabbreviating expressions (beginning on page 1-37 
of Unit 1) we can introduce a tool that can be used in more complex 
situations and we can give the students the necessary practice in using 
it. This tool is the notion of principal operator . 

When all the grouping symbols have been restored in unabbreviating 
an expression the principal operator is the one which corresponds with 
(or links) the pair of outermost grouping symbols. For example, the 
principal operator in '3 + 4 X 5' is *+' because unabbreviating this 
expression gives us: 

{3 + (4 X 5)} 
The principail operator in: 

{([3 X 3] + [5 X 3]) - 10} 




is '-'. It naay be helpful not to omit the outermost grouping synabols as 
quickly as we have been doing in the past. 

[An expression whose principal operator is '+' is sometimes called 
*an indicated sum', oneAvhose principal operator is 'X' is called 'ctn 
indicated product'. ] Before assigning the exercises on page 1-40 we 
might ask the class to name the principal operator in each of several 
exercises. A student who is able to do this immediately in an exercise 
such as: 

3+8-2+5, 

5X6 + 3X4, 
and: 

4 + [5 - 3 - 1] X [7 + (2 X 4)] 

must be using the conventions correctly. 

There are several places in the first four units where the tool, 

principal operator, may be used to advantage. One of these is discussed 

in the following pages. During the year, we will bring others to your 

attention. 

o, o^ o, 

'1^ '4^ 'I* 



i.:i. 



PATTERN -SENTENCES, INSTANCES, CONSEQUENCES 



Distinguishing between 

sentences which are instances of a principle 
[and, so, are also consequences of that principle] 
and 

sentences which are consequences of a principle 
ajid are not instances of that principle 

is often quite difficult. 

"We hope that the suggestions which follow will help resolve this 
difficulty. 

Following page 1-45 of Unit 1, arrange to have: 



^^ l4.5-H2.'5S-- a.96 -f-4-.5 



on one section of your blackboard. 

Now begin sonaething like this: 

Teacher: Write an instance of the commutative principle for addition on 
your paper. John, if JNfery has actually written an Instance of 
the commutative principle for addition what is one symbol 
which must be on her paper ? 

[From here on we will picture the blackboard after the question 
has been cuiswered. ] 



or 




Teacher: What other symbols 
must she have? 



- + -I- 



•i:^:;;: f: 



- 5 - 



[We hope you have to change the above,] 



+ - 4- 



Teacher: What else does she have? 

Student: Numerals. 

Teacher: Where will one of these numerals be written? 

Student: Before the first '+'. 

Teacher: Here? [pointing and drawing the * '. 




Student: A copy of that numeral must be written after the second *+*, 

Teacher: Let's indicate that we want a copy of that numeral by using 
another ' *. 



+ - -f 



Teacher: What else must Mary have? 

Student: A numeral after the first *+'. 

Teacher: Does that have to be a copy of the numeral we placed before 
the first *+'? [No!] Let's indicate that by using a *maj\ 




[Continue until you have a pattern -sentence] 



! 1 ' ■■ •'• ." ■• '■' '■ i'"i -I 



•).:j «••!; 



j^ . . >■' Y** i^t'''!" ^■^' "' 



•11 I /. 






ft"? vli,0 '• 






I .-.: 



■.' '; •■! 



6 - 




Teacher: We call this a pattern -sentence for the commutative principle 
for addition. If you have actually written an instance of the 
connmutative principle for addition, then if we put the proper 
numerals in the blanks in this pattern -sentence we will have 
a copy of the sentence you wrote. 

Mary, tell me how to fill in these blanks so that we will 
have a copy of your sentence. 

Student: Put a '4' above each ' ' and a '7' above each '/uvu'. 

Teacher: Is this a copy of your sentence? 

Student: Yes. 

7= 7 4- 



Teacher: Then your sentence is an instance of the commutative principle 
for addition. Look at: this sentence. Is this an instamce of the 
commutative principle for addition? 



-"T" /vi/u ~Afyi'' 



5-^7-^3 ^7-+5 + 3 



Let's unabbreviate it. 



-+- 



MA) ATih 



+ . 



{(5+7)+3]4(7^-5)4-3} 



What is the principal operator on the left side of the pattern 
sentence? What is the principal operator on the left side of 
the other sentence ? 



i 



mhj /i/vb 



+- 



Z_ 



; 



((5+7)+3]4(7-hS)+3] 



Teacher: If this is an instance, then we must be able to fill the blanks 
in the pattern sentence so that we get a copy. What shall we 
place above the ^o/jx, ' on the left side of the pattern-sentence? 

Student: A '3*. 

Teacher: Where else must a '3* go? 

Student: Above the '/viaj ' on the right. 

Teacher: What must go above the ' * on the left side? 

Student: A '(5 + 7)' abovfe each ' 



(q-h7) H- 3 -=: 3 -h (5-h7) 



Teacher: Is this a copy of our original sentence? 

Student: No. 

Teacher: Then our original sentence is not an instance of the commu- 
tative principle of addition. However, an instance of the 
comnnutative principle for addition can be used to help us 
decide if such a sentence as: 



(759 + 78) + 95 = (78 + 759) + 95 



is true. 



SUo: (7 59+78)-l'95-(78-l- 759}-f 95 



+ 



AAMj " AA/i/v 



4- 



What is the instance? 



Student: '759+78=78+759'. 



•.V ; i 



sLju>'. (75^ -f 7S)+95 -(75+759)+95 
(75^-v7a) =(78+ 75?) 



Teacher; 



If we accept the commutative principle for addition then with- 
out doing any computation we must believe that '759 + 78' and 
*78 + 759' name the same number. 



^4(xyio ' 


(759 +78)- 


+ 95^(7S + 759) + S5 




(759 ■V7S>) 
(759 + 7&) 


^(7S+ 7 59) 
+ 95 =. 


[cps^ 




/K 



Student: 

Teacher : 

Student: 

Teacher: 

Student: 

Teacher: 



What is the simplest way to complete this sentence so that we 
will have a true statement? 

'95 + (759 + 78)', 

Too hard. 

'932'. 

Too hard. 

'(759 +78) + 95'. 

Right ! ! ! 



5Xno-. (759+78)-^95-(76+759) + 9S 

W (759 +78) = (78+759) L^pa] 

(759 +78) +95^(759 + 78) +9 5 



Teacher: Now how are we going to use (*) to show that the sentence at 
the top of the board is true? 

[Deathly silence ! ! ! But wait . If it doesn't come, go ahead.] 

"What does (*) tell you? (Mean to you?) 

Student: (*) tells me that '(759 + 78)* and '(78 + 759)' are names for 
the same number. 

Teacher: Do you have to use the name '(759 + 78)'? 



9 - 



Student: No. 

Teacher : Do you have to use the name '{78 + 759)'? 

Student: No. 

Teacher: Can you use either name you decide you wauit to use? 

Student: Yes. 

Teacher: Let's look at: 

(759 +78) + 95 = (759 + 78) + 95 

and decide whether we want to use the narne '(759 + 78)' or 
the name '(7 8 + 759)'. "Which one do you want to use? 

Student: Keep a '(759 + 78)' on the left side and use a '(78 + 759)' on 
the right side. 

Teacher: What shall I write? 



Stouji (75^4-78) i-^g-(T84759) + 95 

(759+78) -(76 + 759) L^paJ" 
(l5 9 + 78)+<?5 ^( 759 +78 ) +95 
(7 59 +7S)+95x(78 + 7 59)-f95 




Is this what we wanted? 
Student: Yes . 
Teacher: We usually abbreviate all this as: 




Is this sentence an instance of the cpa? 
Student: No. 

Teacher: Did we use an instance of the cpa? 
Student: Yes. 
Teacher: We say that 

(759 + 78) = (78 + 759) 

is both an instance ajid a consequence of the cpa. 



»;!'• 



10 



Teacher: However, 

(759 + 78) + 95 = (78 + 759) + 95 

is a consequence but is not an instance of the cpa. 

[Obviously, the dialogue above is intended only to indicate 
possible questions and answers. ] 

■J, 

Now develop pattern sentences for the cpm, the apa, and the apni. 

cpm: 

) 



pa: 


(_ 


+ ) + 




_ ■'" ^/VWh'^ - 


or 




_ + (AaAa.+ ----) = 


(_ 


'^AAAfV ' "*" 


or 




_ IX^lAAAJ ~ 




_ "^ ' OAJW'^ . 


or 




— ^WXnAA/ ' 




— nnAA/ 



) 



apm: sinailar to those for the apa. 

[Be certain that the students see that the four pattern -sentences for the 
apa are equivalent. ] 



[The following material is very much condensed.] 

Teacher: Now let's exannine Exercise 14 of A on page 1-48: 

72 + (45 + 63) +85 = 85 + [72 + (45 + 63)] 

We are told this is an instance of one of the four principles. 
First, let's unabbreviated 

{[72 + (45 + 63)] + 85} = {85 +[72 + (45 + 63)]} 

t r 

What is the principal operator in the expression on the left side? 

Student: The third *+'. 

Teacher: In the expression on the right side? 

Student: The fourth '+'. [The first '+' on the right side.] 

Teacher: These '+' signs must match the principal operators in the two 

sides of one of the patterns. Which pattern do you want to try? 

Student: + 85 = 85 + 

• . A/lAO iVWX. 



«yoVj 



11 



f 72 + (45 + 63) 1 +85 = 85 + [7 2 + (45 + 63)] 

.t/'la/v nnrvx 

Teacher: Is this a copy of the unabbreviated sentence? 

Student: Yes. 

Teacher: Then the original sentence is an instcuice of the cpa. 

Now look at Exercise 13, and unabbreviate. What is the 
principal operator on the first side? On the second side? 

{(72 + 45) + (63 + 85)} = {72 + [45 + (63 + 85)]} 
(72 + 45) + (63 + 85) = (63 + 85) + (72 + 45) 

/I^IA/WVU O/VlAAvVlA 

Teacher: Is this a copy? 

Student: No. 

Teacher: + ( + ) = ( +^„„ ) + 

(72 + 45) +(.„„., + )= f(7 2 + 45) +„„^„ \+ 

(72 + 45) + ( 63 + 85 ) = ((72 + 45) + 63 ] + 85 

Is this a copy? 

Student: No. / 

Teacher: Let's see if the second side of our sentence will fit the first 
side of the pattern: 

Ui ■*" ( nnAAA "^ ) ~ ^lA ■*"/vwiA. ) "^ 

72 + (45^ + (6_3_ + _8_5.)) = (12 +^^^^) + (63_+_85) 

Is this a copy of our sentence ? 

Student: Not quite, 

[Now we can emphasize that if we believe that '72 +[45 + (63 + 85)]' and 
'(72 + 45) + (63 +85)' name the same number, then we should believe that 
* (7 2 +45) + (63 +85)' and '7 2 + [45 +(63 + 85)]' name the same number. Pay- 
ing attention to the symmetric property of equality at this time will help 
in the proofs later. ] 

Teacher: So the sentence in Exercise 13 is an instance of the apa. 

We hope that this development using the tool, principal operator, 
will be helpful in the discussion of instances and consequences. 



. J. 



-«■<<! i • ' ,-lJ., 



[[{vy : L-) i- c: 



.Tl 



r . •■. n:; :•:-. 



=-<->.■-> < 



.>i; 



//c!; 



J'o) -1- <l*i r 

. .C/'li- J. 



ii-rr.-iO'j Ji'.i 



UICSM-NETRC Tests 

A series of 18 short tests on Units 1, 2, and 3 has been prepared for use 
in the teacher -training experiment sponsored by UICSM and the National 
Educational Television and Radio Center. All are timed and of the "objective" 
type. Most are a little speeded. We present here the first four of these, 
though they probably cannot be used until next year by most teachers. The 
tests are lettered serially from A through R and the number of the page of 
the text which should have been covered before the test is adnninistered is 
given in brackets immediately after the letter. UICSM teachers should feel 
free to use any or all of these items in their tests and quizzes and to send us 
improved versions of any that seenn unclear or otherwise defective. The 
Newsletter staff will of course be pleased to receive any new and ingenious 
items devised by teachers who wouldn't mind sharing thenn with other UICSM 
teachers. --R.S. 

Test A [1-7] (10 minutes) 

Directions : Mark 'A' if the sentence is true. Mark 'B' if it is not true. 

1. Most people would be frightened if they found 'a live cobra' in their 
desks. 

2. 'Alaska' = 'the largest state in the United States' 

3. George Washington = the first president of the United States 

4. '(13 + 3) ^ 4' is a name for '6-2'. 

5. The word 'ambiguous' is printed below: 

ambiguous 

6. '3' is not a number. 

7. 17-3 is the same as 2 X 7. 

8. If told to write a numeral for 5, you would be correct if you wrote: 

'4+1' 

9. 'Chicago' is a large city. 

10. '10 - 3' is a numeral for 7. 

11. '(3 + 5) X i' is larger than '7'. 

12. is a numeral having an oval shape. 

Assume a single straight "road" for the trips in problems 13 through 24. 

13. If the distance from A to B is twice the distance from B to C, then C is 
farther from A than B is . 



\j- T^llp•' 



:-»-'Vi»- a,<' 



13 - 



14. If a man takes a trip of 20 miles, then another trip of 20 miles, he will 
then be 40 miles frora his original starting point. 

15. If two people each take a trip measured by the same real nunnber, they 
will then be just as far apart as they were when they started. 

16. If two people each take a trip of one mile, they will then be just as far 
apart as they were when they started. 

17. If C is north of A and also north of B, then B is north of A. 

18. If the trip fromi A to B is measured by the same real number as the trip 
fronn C to D, then the trip from A to C is measured by the same real 
number as the trip from B to D. 

19. If the result of several successive directed trips is a return to the original 
starting point, some of the trips were in opposite directions. 

20. Numbers of arithmetic are not used to nneasure trips because they do not 
tell how long each trip is. 

21. After the unit and positive direction have been chosen, each possible trip 
is measured by just one real nunnber. 

22. After the unit and positive direction have been chosen, each real number 
measures just one trip. 

23. If the trip from M to L is measured by the same real number as the trip 
from N to L, then N is twice as far frona M as L is. 

24. Suppose you are to take a trip from the starting point of the first to the 
ending point of the last of the successive trips measured by sonne given 
real numbers. Then no matter which direction was chosen as negative, 
this trip will be measured by the same real number. 

Key for Test A [1-7]: 

1. B 2. B 

7. A 8. B 

13. B 14. B 

19. A 20. B 

Test B [1-15] (15 minutes) 

I. Choose the correct simplification: 

1. "13 + ''3 
(A) no (B) -10 (C) ^6 (D) -16 

2. *6 + "8 
(A) *14 (B) "14 (C) *2 (D) "2 



3. 


A 


4. 


B 


5. 


A 


6. A 


9. 


B 


10. 


A 


11. 


A 


12. B 


15. 


A 


16. 


B 


17. 


B 


18. A 


21. 


A 


22. 


B 


23. 


B 


24. A 



c. 



iHilifi: .1 



"if .'.rr« 



, 'tXiCij -.'-IJ 



or '. i? "■ .". Oii. 



- 1.1 



.0:1 ilufi? ».' 



•iK 



.^■">"i .i >•• 



. ! J ■ ,, A. 









i. •; ftn: 






J 



.•9f 



A^W 



r^* (.'Ji) 



•^■ <•.-'. 



14 - 



3. ("2 + *7) + "5 

(A) *14 (B) 

4. CZ + ~7) + "9 

(A) -4 (6)0 



(C) *4 



(C) -16 



(D) -4 



(D) -14 



5. [{*13 + -5) + ""5] + -13 

(A) ns (B) *8 

6. "29.03 + *99.98 

(A) ^70.95 (B) "70.95 

7. ("37 + *25) + "12 

(A) "24 (B) *24 

8. {*231. 57 + "548.59) + "18. 37 
(A) "335.39 (B) "298.65 



(C) "8 



(C) -74 



(D) 



(C) *129.01 (D) "129.01 



(D) 



(C) "266.18 (D) "213.20 



II. Choose the answer which will make the sentence true: 
9. + "3 = "20 





(A) *23 (B) "23 


(C) 


"17 (D) 


*17 


10. 


"14.8 + *6.9 = + "7.9 










(A) *13.8 (B) 


(C) 


"■15. 8 (D) 


*1 


11. 


(*3 + ) + "5 = "3 










(A) *5 (B) "8 


(C) 


-2 (D) 


"1 


12. 


+ "3= +-(24-^6) 










(A) '4' in both blanks 


(B) 


'"4' in both blanks 






(C) '0' in both blanks 


(D) 


none of these 




13. 


(*7800 + -85) + = *85 










(A) "7630 (B) -7715 


(C) 


"7800 (D) 


"7885 


14. 


19 - 3) = + -3 










(A) "3 (B) "6 


(C) 


9 (D) 


12 


15. 


"[(10 - 7) + 15] = ("10 + "7) + 









(A) "1 . (B) ^5 



(C) "15 



(D) *35 



{.-'" !i^.),' 



- 15 



16. "(27 - 3) + 



= { 



+ ~37) + *25 



(A) '0' in the first blank and '*12' in the second 

(B) '*4' in the first blank and '"8' in the second 

(C) '"12' in both blanks 

(D) none of these 

III. Choose the correct answer to the question: 

17. What is the sum of a negative number and a negative number? 
(A) a positive nunnber (B) a negative number 
(C) (D) cannot tell 

18. What is the sum of a positive number and a negative number? 
(A) a positive number (B) a negative numiber 
(C) (D) cannot tell 

I 

19. If the sum of a positive number and another number is a negative 
number, what was added to the positive number? 



(A) a positive number 
(C) 



(B) a negative number 
(D) cannot tell 



20. If the sum of a negative ^number and another number is zero, what 
was added to the negative nunnber ? 

(A) a positive number (B) a negative number 

(C) (D) cannot tell 

Key for Test B [1-15]: 



1. 


B 


2. 


D 


3. 


B 


4. 


D 


5. 


D 


6. 


A 


7. 


A 


8. 


B 


9. 


C 


10. 


B 


11. 


D 


12. 


D 


13. 


A 


14. 


A 


15. 


A 


16. 


B 


17. 


B 


18. 


D 


19. 


B 


20. 


A 







Test C [1-32] (15 minutes) 

Be careful not to confuse addition signs with multiplications signs. 

I. Choose the correct simplification: 

1 . 6 X "8 

(A) 48 (B) "48 (C) "2 (D) 14 



"3X5 
(A) 2 



(B) 15 



(C) "15 



(D) -8 



-c d:-r':.; ;' 



- 16 - 



3. "7 X "8 
(A) -15 

4. {3 + -5) X 6 
(A) 4 



(B) "56 



(B) 12 



5. {"5 X "6) + -30 

(A) "900 (B) "60 

6. 18 + ("6 X -3) 

(A) 36 (B) 9 



7. (-13 X 4) X-{^) 

(A) ^ (B) 4 



8. [{-3 X 2) + (-1 X -6)] X'-9 
(A) (B) 9 



(C) 56 



(C) -12 



(C) 



(C) 324 



(C) -4 



(C) 108 



(D) -1 



(D) "90 



(D) 60 



(D) 



(D) -(f|) 



(D) "108 



II» Choose the answer which will make the sentence true; 
9. "7 X 



= -56 

(A) 63 (B) -49 

10. "14 X = (7 + "10) + {-3 X "1) 



(C) -8 



14. (2 X ^1 + 1 = -9 

(A) -12 (B) -5 



15. 2 X 



3 X 



(A) '0' in each blank 
(C) '-6' in each blank 



(C) -4 



(D) 8 



(A) J 


(B) .-(y) 


(C) 


1 
2 


(D) 





11. -4 X 


-" (B. i 


(C) 


4 


(D) 




(A) -.25 


5 


12. 4 X 


= "6X8 

(B) -12 


(C) 


2 


(D) 




(A) 16 


~2 


13. (5 + 


) X -3 = 3 










(A) -6 


(B) -4 


(C) 


-(i) 


(D) 


6 

5 



(D) 4 



(B) '6' in each blank 
(D) none of these 



^"■- ibi 



4- /.i 



1/ 



16. (3X ^1 + 2= +{2X ) 

(A) '1' in each blank (B) 'T in each blank 

(C) '0' in each blank (D) none of these 

III. Choose the correct answer to the question: 

17. What is the product of a positive nunaber and a negative number? 
(A) a positive number (B) a negative number 

(C) (D) cannot tell 

18. If the product of zero and a real number is zero, by what was zero 
multiplied? 

(A) a positive number (B) a negative number 

(C) (D) cannot tell 

19. If the product of 1 and a real nunnber is the real number, by what 
was 1 multiplied? 

(A) a positive number (B) a negative number 

(C) (D) cannot tell 

20. If the product of a real nunnber and a real nunnber is zero, what can 
you conclude about the real nunnbers ? 

(A) both are nonnegative (B) both are zero 

(C) at least one of thenn is zero (D) nothing 

Key for Test C [1-32]: 



1. 


B 


2. 


C 


3. 


C 


4. 


C 


5, 


C 


6. 


A 


7. 


B 


8. 


A 


9. 


D 


10. 


D 


11. 


A 


12. 


B 


13. 


A 


14. 


B 


5. 


A 


16. 


D 


17. 


B 


IS. 


D 


19. 


D 


20. 


C 







Test D [1-59] (15 minutes) 

Only numbers of arithmetic are used in this test. 

I. Choose the answer that will make the sentence true. 

1. 7X2 + 7X = (2 + 3) X 7 

(A) (B) 2 (C) 3 (D) 5 

1 5 

2. 5 X 34- = + T 

(A) 3 (B) 5 (C) 10 (D) 15 



18 



3. 7X8 + 5X8 = 40+ 

(A) 5 (B) 6 



X 8 



4. 7 X 13 + 13 X 
(A) 13 



= 13X13 



(B) 10 



5. (5 + 



(A) 3 



6. 13 X 



) X 5 = 25 + 40 
(B) 5 



X 7 = 100 



(A) '4' in both blanks 
(C) '6' in both blanks 

7. 987 X 593 + 13 X 593 = 

(A) 98,700 (B) 58,700 



8. 4X 3 + 4 X 



4 X 



+ 15 



(A) '3' in both blanks 
(C) '5' in both blanks 

9. 13 + 2X11 = 13X2 + 

(A) 7 (B) 9 

10. 139 X 672 + 139 X 328 = 

(A) 139,000 (B) 13,900 



(C) 7 



(C) 7 



(C) 8 



(D) 8 



(D) 6 



(D) 11 



(B) '5' in both blanks 
(D) none of these 



(C) 59,300 



(D) 593,000 



(B) '4' in both blanks 
(D) none of these 



(C) 11 



(C) 46,700 



(D) 143 



(D) 146,700 



II. Choose the principle which justifies the given sentence. 

11. 7+2X3 = 7 + 3X2 

(A) apm (B) cpa (C) cpm (D) apa 



12. = 19 X 

(A) apm (B) cpm (C) pnaO 

13. 5 + 3 + 8 = 5 + (3 + 8) 

(A) cpa (B) cpm ( C) apa 

14. (47 + 3) X 9 = [(47 + 3) + 0] X 9 

(A) cpm (B) apnn ( C) apa 



(D) paO 



(E) dpma 



(E) Mpma 



(D) apm (E) dpma 



(D) pmO (E) paO 



19 



15. (3 + 14) X 7 = (14 + 3) X 7 

(A) cpm (B) cpa (C) apa (D) dpma (E) idpma 

16. 4 X (3 + 19) = 4 X 3 + 4 X 19 

(A) apm (B) dpma (C) idpma (D) cpm (E) apa 

17. (48 + 19) X 1 + 12 = 48 + 19 + 12 

(A) dpma (B) idpma ( C) apm (D) pml (E) apa 

18. 37 X 1 X (12 - 3) X 7 = 37 X 1 X [(12 - 3) X 7] 

(A) apm (B) cpm (C) apa (D) cpa (E) pml 

19. 5 X l|- = 5 X 1 + 5 X I 

(A) pml (B) apm (C) cpm (D) idpma (E) dpma 

20. (y + y) X 6 = J X 6 +, i X 6 

(A) dpma (B) idpma ( C) apa (D) apm (E) pml 

III. Choose the correct answer. 

21 . Which of the following is an instance of apm? 

(A) (72 X 45 X 63) X (85 X 22) = (72 X 45) X [63 X (85 X 22)] 

(B) 71 X (52 X 13) = (52 X 13) X 71 (C) 5 X 3 X 4 = (5 X 3) X 4 
(D) 23 X (5 X 1) = 23 X 1 X 5 (E) none 6f them 

22. Which of the following is a consequence of just apa? 

(A) 2 + 3X4=2X(3 + 4) 

(B) 5 + 7 + 2 = 7 + (5 + 2) 

(C) [(7 + 4) + 6] + 8 = (7 + 4) + (6 + 8) 

(D) 6 + (8 + 9) = (6 + 9) + 8 

(E) none of them 

23. Which of the following is a consequence of our punctuation convention 
alone ? 

(A) 72 + (45 + 63) + 85 = 72 + (45 + 63 + 85) 

(B) (72 + 45) + (63 + 85) = 72 + 45 + (63 + 85) 

(C) (72 + 45 + 63) + 85 = (72 + 45) + (63 + 85) 

(D) 72 + 45 + (63 + 85) = 72 + [45 + (63 +^85)] 

(E) none of them 



i::-i) 



8:t (• 



■r. •i-,a,£ 



20 



24. Which of the following is an instance of cpm? 

(A) 8X3 + 7=3X8 + 7 

(B) 5 X(4 + 3) = (3 + 4) X 5 

(C) 2X(1 + 3) = 2X1 + 3X2 

(D) 13X5 + 8X5 = 5X13 + 5X8 

(E) none of them 



Key for Test D [1-59]: 



1. 


C 


2. 


D 


3. 


C 


4. 


D 


5. 


C 


6. 


B 


7. 


D 


8. 


D 


9. 


B 


10. 


A 


11. 


C 


12. 


C 


13. 


C 


14. 


E 


15. 


B 


16. 


C 


17. 


D 


18. 


A 


19. 


D 


20. 


A 


21. 


A 


22. 

'1^ "1" 


C 


23. 


B 


24. 


E 



News and Notices 





PLEASE 






send us your correct address, or the address 


to which 


you 


would 


prefer future issues 


o£ this Newsletter 


be sent. 


We 


will al 


so be grateful for th 


2 names and addresses of others 


whom 


you feel should receive copies of this p 


ublication 


. A 


postal 


card will suffice. 

UICSM 
1208 W 
Urbana 


Newsletter 
. Springfield 
Illinois 







Mr. Beberman will be traveling extensively during the next three months 
while visiting cooperating schools and making presentations before various 
professional groups. His itinerary follows. 

October 

10-13 Visit Pine Bluff, Arkansas, schools. 

14-15 Presentation to Arkansas Council of Teachers of Mathematics, 
Arkansas State College, Conway. 

21 Presentation at Western Zone meeting of the mathematics section, 

New York State Teacher's Association, Snyder. 

24-25 Visiting Villa Maria Academy and Cathedral Preparatory School, Erie, 
Pennsylvania. 



■yrti« .*. \ 



i. 



,^i 



- 21 - 

27 Presentation to meeting of School Superintendent's Division, Catholic 

Educational Association, Peoria, Illinois. 

28-29 Mathematics Teacher's Symposium, Villanova University, Villanova, Pa, 

31 ^ 

November f Visit Cheltenham, Pa., public schools. 

1 ^ 

2 Visit Ridgewood, N.J., public schools. 

3 Visit Pascack Valley Regional High School, Hillsdale, N.J, 
10 Visit public schools in Cleveland Heights, Cleveland, Ohio. 
14-16 Visit Pittsburgh, Pa., cooperating schools . 

17-18 Presentation at annual meeting of Edison Foundation, Pittsburgh. 

December 

8 Visit Whittier, California, public schools, 

9-11 Presentation to annual meeting of California mathematics teachers at 
Asilomar . 

13 Presentation to National Science Foundation Institute at Sacramento . 

State College, Sacramento, California. 

30 Presentation at Christmas meeting of the National Council of Teachers 

of Mathematics, Tempe, Arizona. 

Mr. Eugene Epperson of Talawanda High School, Oxford, Ohio, will speak 
at the annual meeting of the Western Ohio Teacher's Association in Dayton on 
October 28. He is also to appear as a teacher-panelist at the Regional Orien- 
tation Conference in Cincinnatti on December 15th. 

Miss Gertrude Hendrix. Project Teacher Coordinator and Programmer 
for NETRC Math Study films, is the author of "The Case for Basic Research 
in the Theory of Instruction" in the May, I960, American Mathematical 
Monthly (vol. 67, pp. 446-7). She also co-authored "The UICSM Teacher 
Training Films" in the August-September, I960, issue of the same journal 
(vol. 67, pp 686-7) with Mr. Byrl Sims, who directed the film unit. 

Five former UICSM students are among the seven from the new Newton 
(Mass.) South High School who passed the qualifying examination of the National 
Merit Scholarship Program. Their names are Ray Frieden, Ralph Pollack, 
Shepard Golub, Philip Alpert, and Laura Cohen. 

Reprints of Mr. Beberman's article on "Improving High School Mathe- 
m.atics Teaching", as it appeared in the December, 1959, Educational 
Leadership , are still available from the project office. The following reprints 
are also still available: 

UICSM Project Staff . Words, 'Words', "Words". 

UICSM Project Staff. Arithmetic with Frames. 

Gertrude Hendrix. Variable Paradox: A Dialog in One Act. 

M. Eleanor McCoy. A Secondary School Mathematics Program. 



v_y i 





ews dVd r 



An occasional publication, of the 
UNIVERSITY OF ILLINOIS COMMITTEE ON SCHOOL MATHEMATICS 

1208 West Springfield 
Urbana, Illinois 



Absolute Valuing 

A Very Short Short -Course 

UICSM-NETRC Tests E-H 

Quiz [4-41] 

News and Notices 



2 

8 
12 
20 
25 



UICSM Newsletter No. 2 



November 9, I960 



: i 



ABSOLUTE VALUING : TWO OPERATIONS , ONE NAME 

Many of the questions asked about material in Course I are inquiries 
about absolute valuing. These questions are closely related to the 
problem raised by using numerals for numbers of arithmetic as names 
for real numbers. This problem would be simplified by a certain 
amount of rewriting in Unit I. If you are interested in these contem- 
plated changes, mail us a card and we'll send you a copy of themi. 

Until changes are made in the text, perhaps some of the confusion con- 
cerning absolute valuing can be eliminated by the following discussion: 



Consider the operations A and B, some of whose ordered pairs are 
named below: 





In these tables numerals for numbers of arithmetic 
are not being used as names for real nunnbers. 



A is an operation which miaps the reals unto the set of numbers of arith- 
metic. B is an operation which nnaps the reals onto the set of nonnegative 
reals. Since these operations contain different ordered pairs, A /^ B. 



Operations A and B are both important. A and its "partial inverses", 
* and ~, are needed in order to get back and forth between the real 
numbers and the numbers of arithmetic --for each real number x, A(x) 
is the number of arithmetic "corresponding" to x, and, for each number 
a of arithmetic, *a and "a are, respectively, the nonnegative real number 
and the nonpositive real number which "correspond" to a. Thus, for 
example, these three operations are used when one adds or multiplies 






qf 



." v.. 



< • . 



t 



•J. .> 



-3 



real numbers by adding, subtracting, or multiplying numbers of arith- 
metic. And, notation for them is needed if, as on pages 2-28 and 2-29, 
one is to state the rules for adding cind multiplying real numbers. 
Another place where we make explicit use of A [and use | | ' as a name 
for a] is in Unit 6. We do so when discussing coordinate systems on 
pages 6-232 and 6-233; in exercises, on pages 6-235, 6-236, and 6-237, 
leading up to the distance formula; and in the distance formula itself, on 
pages 6-238 and 6-239. 

The isomorphism between the system of nonnegative real numbers and 
the system of numbers of arithmetic is expressed by: 

V if X is nonnegative then *A(x) = x, V AC^a) = a 

[domain of 'a' = set of numbers of arithmetic], 

V V if X amd y are nonnegative then x + y = '^(A(x) + A(y)) 

and x'y = *{A{x) • A(y)), 

V V if X and y are nonnegative then x > y if and only if A(x) > A(y) 
x y 

It is this isomorphism which suggests that numerals which have been 
introduced as names for numbers of arithmetic may, on occasion, be 
used as names for nonnegative real numbers [see page 1-31]. [in 
anticipation of this we have, fro-nn the beginning, used the symbols 
'+', 'X', and '>', which originally refer to two operations and a relation 
for numbers of arithmetic, to refer to the "corresponding" operations 
and relation for real numbers. ] It is also the existence of this isomor- 
phism which makes it possible for us to take a further step and "pretend 
that measures [which are numbers of arithmetic] are real numbers" 
([6-35]) and to "think of them "as if they were real numbers" " 
(TC[3-55]). See, also, TC[5-196]. 

The operation A is, as indicated above, importamt in two kinds of situ- 
ations. First, when one is dealing with the foundations of the arithmetic 
of real nunnbers as based on the arithnnetic of the numbers of arithnnetic, 
aiid» -Second, when one is applying theorems proved for real numbers to 
solve problems which are, actually, concerned-with numbers of arithmetic. 






:v fii :i' 



Iff ir. '.•.^■ 



•■ f " -7r ' 



■if-' ;-''C'i': 



..1 X • 



n-.ir 



V 



7 :: 



;? 



•■•n' 



The second kind of situation occurs very frequently --in particular in the 
solution of worded problems in Units 3, 4, and 5, and throughout Unit 6. 
However, explicit reference to A and *is seldom, made in this connection. 
But see, again, TC[5-196] and the references there cited. 

The operation B is of importance in the theory of real numbers itself. 
It can be introduced without reference to numbers of arithmetic either 
by: 

V if X is nonnegative then B(x) =x and if x is nonpositive then B(x) = -x 
or, more succinctly, by: 

V B(x) = 7^2 

Alternatively, B may be defined in termis of A and *: 

V B(x) = *A(x) [For, V if x is nonpositive then *A{x) = -x. ] 

The most frequent use we make of B is to obtain a wider range of exer- 
cises than we otherwise could. There is not much of this in Unit 2. But 
see, especially, the last sentence beginning on TC[2-3, 4, 5]a--this 
should be recalled when doing Part B on page 2-111. There are many 
examples of this use of B in Unit^3. Also, our only two "professional" 
uses of B are in Unit 3, pages 3-118 et seq. , and page 3-132 et seq. 

Of the three possible definition.s of B, noted above, only the first and 
third are pedagogically suitable. Because we have operation A and oper- 
ation "^available, and for another reason, we chose to use the third. 
The other reason is that we follow the custom of using numerals for 
numbers of arithmetic sometimes to nanne numbers of arithmetic and 
sometimes to nanae the corresponding nonnegative real nunnbers. [This 
is an unfortunate custom, but it is so wide -spread that students must 
learn to live with it. ] Because of this custom, a '2*, in a given context, 
may be intended to nanne the number 2 of arithnnetic or it nnay be in- 
tended to nanne the real nunnber *Z. [Of course, only a schizophrenic 
can use the same '2' to name both the number 2 of arithnnetic and the 
real number *2. ] For example, in the expression '*2', the '2' must be 
intended as a name for a number of arithmetic, for the donnain of the 
operation * contains only such nunnbers; cind in the expressions 'A(2)' 



•■jr if 



A\i -J- 



• T 



'11 ; 



and 'B(2)' the '2's cein only be names for *Z, for the domain of either 
A or B contains only real numbers. Now, one consequence of adopting 
this custom is that, having introduced ' | . . . I* as a substitute for 
'A(. . . )', the expression ' 1 *2 | ' [or ' | 2 ] '] which, at the time ' | . ; . | * is 
introduced, is a name for the number 2 of arithmetic, may, in the 
future, be used sometimes as a name for this number and somietimes 
as a name for ^. In the latter case, ' | . . . | ' is being used as a sub- 
stitute for 'B(. . . )', instead of for 'A(. . . )'. So, following the custona 
means, in this connection, that the same symbol '| |' will refer some- 
times to the operation A and sometinnes to the operation B. A somewhat 
similar situation has already occured in connection with ' +', 'X', and 
'< '. For example, '+' refers sometimes to the operation of addition of 
numbers of arithmetic and sometimes to the operation of addition of 
real numbers. The difference in the two situations is that in, for 
example, '2 + 3', one knows which operation '+' refers to as soon as 
one has learned [from the over-all context] which kind of numbers '2' 
and '3' are being used as names for. But, in the case of '|2|' one knows 
that, in any case, the '2' is being used as a name for *Z, but one has to 
look at the over -all context to determine which operation the ' | | ' refers 
to. 

Now, ever since the custom we have been discussing was first introduced, 
on pages 1-31 and 1-3 2, we have acted according to the convention that 
numerals which may be used to name nunnbers of arithmetic will, in fact, 
be interpreted as naming the corresponding nonnegative real numbers 
whenever the context does not, either explicitly or implicitly, prohibit 
this interpretation. So, from Unit 2 on, '| |' refers to the operation B 
unless the context prohibits this interpretation. Thus, the sentence: 

1-5| > 2 
is to be considered as a sentence aixjut real numbers and is equivalent to; 

""5 > -^ 
However, the sentence; 

-5 +^ = -(|-5| - |^2|) 

is meajiingless if we think of ' | ~5 | ' as a name for *5 since nonpositiving 
is an operation on numbers of arithmetic. So here we must think of 



.'. ;"1 •■?. ir 



■ / ■' 



:o -.' 



'1.1 






V '.I'l; . / 



.■ :?{ i; 
;r. : f. 

.^..- I-- 






• i" " ';>;;■■ ■ .. -vj-o. j'-^j- :■ 

.-■Jit. .1^ ■ 



.•/:• 



i";t- 



:•|■^.• 



^J 



6- 



' I "5 I ' and ' | '^2 ] ' as names for numbers of arithmetic . 

For another example, in Exercise 9, on page 2-4, the context does not 
prohibit [it even suggests], the interpretation of ' | | ' as B. So, this is 
the interpretation to be given to ' | | ' in this exercise. On the other 
hand, on page 2-29, the context, which includes the last three lines on 
page 2-28, implicitly prohibits the interpretation of the two occurrences 
of ' I I ' on line 5 as references to B [and the phrase 'which corresponds 
with' naight be interpreted as explicitly prohibiting this interpretation]. 
Also, the ' "' in line 12 explicitly prohibits the interpretation of the two 
'I I s on this line as references to B. So, all four of these occurrences 
of I I must be taken as referring to A. In D, 1(a), on page 2-29, the 
last two 'I I 's must, for the same reason, refer to A. There nn.ay be 
some doubt about the first two. Certainly, either both refer to A or both 
refer to B. Because of the isomorphism, with respect to >, between 
the numbers of arithmetic and the nonnegative real numbers, the truth 
or falsity of a statement of the fornn |x| > |y | ' is unaffected by the 
choice of either interpretation. However, the over-all context- -defining 
the arithmetic of the real numbers in terms of that of the numbers of 
arithmetic -would probably be taken as prohibiting the interpretation of 
these 'I I 's as referring to B. 

For another example, in Unit 6, on page 6-23 2, since [whatever it may 
be convenient to pretend] measures are numbers of arithmetic, the 
'I I 's occurring in the displayed sentences must refer to the operation 
A--for, for any points O, L, and U, the naeasures of OL and OU are 
numbers of arithmetic. So, here, and in the pages which follow, the 
context implicitly prohibits the interpretation of ' | | ' as referring to B. 

In summary: 

1. (a) It is customary to use those numerals which may name numbers 
of arithmetic in two ways- -sometimes as names for numbers of 
arithmetic, and sometimes as names for the corresponding non- 
negative real nunabers. 

(b) Numerals in which the principal operator is ' | | ' are of this kind. 
So, we use ' | | ' sometimes for 'A(. . .)', and sometimes for "B{. , .)'. 



'• ;.VjM 



,-.. iJ 



■rna-. 



2. (a) We adopt the convention that, after page 1-32 such "ambiguous 

nunnierails" will, in fact, be interpreted as naming the corres- 
ponding nonnegative real numbers whenever the context does not, 
explicitly or implicitly, prohibit this interpretation. 

(b) So, in Unit 2 and later units, ' | | ' refers to the operation B un- 
less the context prohibits this interpretation. In Unit 1, from 
page 1 -104 through the first half of page 1-110, the text specifies 
that each occurrence of ' | | ' refers to the operation A. The last 
hcdf of page 1-110 calls attention to the possibility of interpreting 
'I I ' as a name for B, It would probably be a help to students if 
the discussion on page 1-110 were extended as follows: 

From now on, in most of your work with absolute values you 
will want to use the real number interpretation. So, let's 
agree that: 

Numerals which contain a '| | ' should be interpreted as 
numerals for real number s - -except in places where the 
context prohibits this interpretation. 

3, Since the same symbol ' | I ' is used to name both operation A and 
operation B, it is natural to use the same words, 'absolute valuing* 
to name both operations , and, in all cases, to read ' 1 • . . | ' as 'the 
absolute value of . . . '. So the meaning of each occurrence of the 
words 'absolute value' must, like the interpretation of ' | | ' be 
determined from the context in which it occurs. As a matter of 
fact, this has been the case traditionadly. Texts used in high 
schools often give rules similar to the following: 

To add two nunabers with the same sign, add their absolute 
values and prefix the common sign. 

Here, the words 'absolute value' refer to operation A. But, when 
a student gets to advanced work he is given the following definition: 

^x> •''I = ^ ^^^ ^x< 1^1 = -^' 

and is told to read '| | ' as 'the absolute value of. So, here, he is 
to use 'absolute value' to refer to operation B.--A, H. 



■ V- ;:• 



;i;'.'.-^' 



8 - 



A VERY SHORT SHORT -COURSE 

Many teachers of UICSM materials ai e receiving requests to conduct 
inservice training courses for teachers. In some cases the time 
allotted for this training is extremely short. A request for advice as 
to what to do in such a situation was recently received here. Our 
response to this request is embodied in the paragraphs which follow. 

You have been asked to conduct a training course for those persons 
who are teaching or will teach the first four units in your school ! 

You are attenapting an impossible task if you try to really teach the 
four units in the time allotted. You may cover the pages but you 
won't uncover much. I would select several topics and ignore the 
rest except for questions that are raised. Since so very many of 
the principles which are used in the last three units are foreshadowed 
in the first unit, I suggest that you take particular care with Unit 1. 
Remennber that the fill-in exercises are very important in clarifying 
the principles. 

You will notice that the sample outline which follows suggests that 
at least three sessions be devoted to Unit 1 alone. Since your course 
is of such short duration, you should insist that each person who 
will take the course have a copy of Unit 1 before the first session. 
If the participants read the Introduction of Unit 1 prior to the first 
session, you will be able to get into the "heart" of the unit sooner. 

Session I 

1. Read the Introduction [if necessary] and discuss the exercises. 

2. Do not use time here for a long discussion of the value of the 
Introduction. Bring this into later discussions of topics 
where misunderstanding of written material could occur if 
we did not have a way of distinguishing between a symbol and 
its referent. 

3. Emphasize that no naore than two days should be spent on 
the Introduction with a class of 8th or 9th grade students. 

4. Assign the first half of Unit 1 for the next session. Be sure 
to tell your group that they should not expect to grasp all the 
material in the TC in one reading. 

Session II 

1. Develop the conventions we use for abbreviating and onab- 
breviating expressions. 

2. Discuss the idea of the principal operator in an expression. 

3. Develop pattern sentences for the basic principles. [The first 
Newsletter contains material that should help on principal 
operator and pattern -sentence.] 



i 






t V * 



iff 



■■".' 



-9 



4. Be sure the teachers understand that 

'(2 +7) +3' and '2+7 +3' 

name the same number because of an abbreviating convention. 
On the other hand, the expressions 

'(2 + 7) + 3' [or, '2 + 7 + 3'] 

and '2 + (7 + 3)' 

name the same number by virtue of the associative principle 
for addition. 

5. Point out that 

'(8 ^ 4) -f 2' and '8 -^ 4 -f 2' 

name the same number by convention, but that 

'(8 -f 4) 4- 2' [or. '8 4-4 + 2'] 

and '8 + (4 + 2)' 

do NOT name the same number. [There is no associative 
principle for division.] 

6. Assign the last half of Unit 1 for the next session. 

Session III 

1. Distinguish between sentences which are instances of the 
principles, and those which are consequences but not in- 
stances. [Of course, any instance is a consequence. ] 

2. Study derivations of such sentences as: 

98 X 20 + 9 X 98 + 98 X 21 = (21 + 9 + 20) X 98 

3. It is wise to do a careful job with derivations of the kind 
naentioned in Item 2 above, since this will make the work in 
Unit 2 much easier. 

4. Assign the first half of Unit 2 for the fourth session. 

Session IV 

1. Discuss operations and their inverses^ [Point out that certain 
operations do not have inverses.] Be sure to include the oper- 
ations opposition, nonpositiving, nonnegativing, and both 
absolute valuing operations in this discussion, [See the first 
article in this newsletter for a discussion of the two operations 
each of which is nanned 'absolute valuing'.] 

2. Stress the manner in which open sentences can be used to 
generate other sentences, and the way pronunaeral expressions 
enable one to generate other expressions. 

3. Page 2-21 usually requires careful attentipn. The use of the 
principal operator is helpful here. 



•i^'i :i'iq 



..d- 



-■'\ih:'}<: 



sc . 



I- I 



- 10 



4. Discuss the pattern for testing statements of the forms: 

(3x + 7y) + (5x + 6y) = (3 + 5)x + (7 + 6)y 
and: 

(3x)(7x) = (3-7)(xx) 

Make certain that teachers understand the role the test -pattern 
plays in a proof. 

5, Assign theorems to be proved. 

Session V 

1. By this time you should be far enough behind that you will need 
this session to catch up with the material outlined for the first 
four sessions. You may want to use the next session this way 
also. 

2. Assign the last half of Unit Z. 

Session VI 

1. Work on proofs. 

2. Assign more theorems to be proved. Have both Unit 2 and 
Unit 3 brought to class for Session VII. 

Session VII 

1. Develop the intuitive approach to solving equations and in- 
equations. ^ 

2. Introduce the concepts of solution sets, loci, and the use of 
brace notation for naming sets. 

3. Emphasize that at this time the student should have no formal 
rules for solving equations and inequations. 

4. Get the teachers involved in solving: 

|xj = 2 
|x| < 2 
|x| > 2 

intuitively. See that many different names are given for each 
solution set. Include the graphing of the solution set. Thus: 

{x: |x| < 2} = "2, 2 = {x: "2 < x < 2} = {x: x^ < 4} 



-^ 3 ^ ^ i E- 

-3-2-1 1 2 



We hope to have an article in the next newsletter which will 
help here. 



.<.■!. ■<; » - )''■ .<J ". il;j 



. . . i; ; 1 ■ ■: , 









liV.- 



-•■I- 



- 11 



5. Be certain that the use of 'and' and 'or' is clear. Use 
colored chalk. Your blackboard may look like this: 






[r- 



rczd or c^r&ers 



s. 




3 M- 5 



6. Assign the first heilf of Unit 3, 

Session VIII 

1. Devote this session to the transformation principles for 
equations. A comparison of "transposition" and addition 
transfornnation principle may be profitable, 

2. Stress the importance of the factoring transformation principle 
■ and its use in solving quadratics by completing the square. 

3. Assign the last half of Unit 3. 

Session IX 

1. Answer any questions about the last half of Unit 3. 

2. Start the discussion of ordered pairs and graphing. Graph: 



{(x, y): X > 2}; 



{{x, y): y < 3}; 



{(x, y): X > 2 and y < 3}; {(x, y): x > 2 or y < 3 }. 

3. Assign Unit 4. 

Session X 

1. Discuss any material about which questions are raised. 

2. Omit Unit 4 completely if necessary. 



Good luck with your in-service courses. We hope this outline will be 
helpful.- -A. H. 



- '.-u .Ij 



; : ./: 



.1- . .Ja, 



«.«.•»' 






■ • ■ .\,j '■ 1 1' i! J5.. 






■ I 



• i. •>■•) .« 



.*•-.•■■ ''i: •( ■.I'll'. '. Kii./'H'.iJ' p ■..■{ /V 



( c '-.ii;! ;jK. .••. :.ij rm 



- 12 



UICSM-NETRC Math Study Tests 



In Newsletter No, 1 we presented the first four tests in the UICSM- 
NETRC series. Here are the next four, designated by the letters E, F, G, 
and H. The page to have been completed is again in brackets after the 
letter. A fifteen minute time limit has been set for each for the purposes 
of the NETRC Math Study. --R.S„ 

Test E [1-7 2] 

1. The operation multiplying by 2 does not contain: 

(A) (1, 2). (B) (0, 1) (C) (2, 4) (D) (^ 1) 



2. The inverse of adding 5 contains: 

(A) (1, 5) (B) (5, 5) (C) (5, 0) (D) (0, 5) 

3. Another name for the inverse of adding is: 

(A) multiplying by (B) subtracting 1 

(C) multiplying by 1 (D) none of these 

4. Which pair belongs both, to adding 3 and to adding 4? 

(A) (3,4) (B) (1,8) (C) (4,7) (D) none of these 

4 

5. Which pair belongs both to multiplying by 2 and to adding 5 ? 

(A) (5,10) (B) (1,7) (C) (10,5) (D) none of these 

6. Which pair belongs both to multiplying by 3 and to the inverse of 
multiplying by 3 ? 

(A) (0,3) (B) (0,0) (C) (1, 1) (D) none of these 

7. Which pair belongs both to adding 4 and to the inverse of adding 4? 

(A) (0, 4) (B) (4, 0) (C) (0, 0) (D) none of these 

8. Which pair belongs both to the inverse of multiplying by 5 and to the 
inverse of adding 12? 

(A) (15,3) (B) (3, 15) (C) (60,0) (D) none of these 

9. The pair (18, 25) belongs to an adding operation. Which pair belongs to 
its inverse ? 

(A) (16,9) (B) (8, 15) (C) (0, 0) (D) none of these 



U. :-a< 



(.i.\ Dn.'A I 



((. 



- 13 - 



10. Which of these operations is not the same as dividing by 5 ? 

(A) the inverse of multiplying by 5 

(B) the inverse of multiplying by the reciprocal of 5 

(C) multiplying by the reciprocal of 5 

(D) the inverse of dividing by the reciprocal of 5 

11. Which of these is a true principle for numbers of arithmetic? 

(A) commutative principle for division 

(B) associative principle for division 

(C) distributive principle for division over division 

(D) none of them 

12. Which of these is not a true principle for numbers of arithmetic? 

(A) commutative principle for addition 

(B) associative principle for addition 

(C) distributive principle for addition over addition 

(D) principle for adding zero 

Choose the principle for real nunnbers which justifies each sentence. 

13. "18 X X 47 + 1 = X 47 + 1 

(A) pml (B) pmO (C) paO (D) apm 

14. "18 X 47 X + 1 = "18 X (47 X 0) + 1 

(A) apm (B) pmO (C) pml (D) paO 

15. (1 + ^) X "8 + 2 X "8 = (1 + "7 + 2) X "8 

(A) cpa (B) apa (C) dpma (D) apm 

16. n + "3) X ("9 + 10 + "2 + 6) = (7 + "3) X [("9 + 10) + ("2 + 6)] 

(A) apa (B) cpa (C) cpm (D) apm 

17. ~1 X 1 X + = 1 X "1 X + 

(A) pmO (B) pml (C) cpm (D) paO 

18. 1 X (0 + 1) X = (1 X + 1 X 1) X 

(A) pml (B) idpma (C) pmO (D) paO 



i',;V»^ 



:• '!'.■ :. 






r.- ■ : I 



-<^'.fy' 



•-\ 



;)...M <;. 



- 14 



19. (0 + X 1) X 1 + = (0 + 0) X 1 + 

(A) apm (B) paO (C) pmO (D) pml 

20. (OXl+0)Xl+0 = OXlXl+0 

(A) apm (B) pml (C) pmO (D) paO 

Key for Test E [1-7 2] 

1. B 2. C 3. C 4. D 5. A 6. B 7. D 

8. A 9. A 10. B 11. D 12. C 13. B 14. A 

15. C 16. A 17. C 18. B 19. D 20. D 

Test F [1-92] 

I^ Choose the answer which will make the sentence true. 
1. -(3 - 8) = 



(A) 5 (B) -5 (C) 11 (D) -11 



2. 3 



(A) -9 (B) 9 (C) -3 (D) 3 



3. -13 - -7 = 



(A) -6 (B)' 6 (C) -20 (D) 20 



4. 4-9-7+13 = 



(A) 1 (B) -1 (C) -7 (D) none of these 

5. 5 - -8 - 2 + -3 - 1 = 



(A) -9 (B) 7 (C) 13 (D) none of these 

6. -(8 + ) = 10 

(A) -18 (B) 18 (C) -2 (D) 2 

7. -{7 - ) = 3 

(A) -10 (B) 10 (C) -4 (D) 4 

8. -(5 + ) -8 = 

(A) 3 (B) -3 (C) 13 (D) none of these 

9. (2 - ) X -5 = -40 

(A) 10 (B) -10 (C) -6 (D) none of these 



;-i;wO: 



' --.I : 



■jl'i I I'i'li' 



1) 



K; .'•■ ^rt-.' 



Vcl 



.) . 



15 - 



10. 3 - X (4 - 10) = 15 

(A) ~ (B) -18 (C) 2 (D) none of these 



II. Choose the correct answer, 

11. Which number is the opposite of itself? 

(A) 1 (B) "1 (C) (D) none of these 

12. Which kind of problem does not always have a solution? 

(A) subtracting numbers of arithmetic (B) subtracting real numbers 
(C) adding numbers of arithmetic (D) adding real numbers 

13. The inverse of adding the opposite of 5 is the same as 

(A) adding 5 (B) subtracting 5 (C) adding "5 (D) none of these 

14. Subtracting the opposite of ~8 is not the same as 

(A) adding the opposite of 8 (B) adding ~8 

(C) subtracting 8 (D) subtracting ~8 

15. If a first real number isvthe opposite of a second real number, then 

(A) their difference is nonnegative (B) their product is nonpositive 
(C) their quotient is 1 (D) none of these 

16. Pick a real number, then add it to its opposite, then nnultiply the sum 
by the number picked. What kind of real number will the product be? 

(A) positive (B) negative (C) (D) cemnot tell 

17. Which operation is the inverse of oppositing? 

(A) oppositing (B) adding (C) multiplying by (D) none of these 

18. A first number is not necessarily the opposite of a second when 

(A) the opposite of the first number is the opposite of the second 

(B) the sum of the numbers is 

(C) the first number is the product of the second number by ~1 

(D) the second number is the product of the first number by ~ 1 



)j. 



■.: iC u I '..II 



' . ■ •■•-) 



•■iT.~.\\ to ; i-:o. 



<..■ \.\i ■•:•, -^ 



•<:• -i' .•■•;.il.=0'j. 



MO L- .. '<.l 



ti TO y '<)i 



I •••>•; •;. . 



Jc. 



.if. 



•LO .••)<ii "s-, .>n ; .',;.: >V 



oi;. J «■. 



■ >■•'. m: 



C'. 



'?C 



J ■ •'*a--.r7",> 



ilJ'C- 



/crM" ' ,'•: 






.-<'- 



M * . .J 



16- 



19. The opposite of the sum of a first number eind a second number is 

(A) the first number nninus the second number 

(B) the second number minus the first number 

(C) the sun:i of their opposites 

(D) the opposite of the first number, plus the second number 

20. After a first number and a second number have been chosen, which 
of these is not always the same as the others ? 

(A) the first number minus the opposite of the opposite of the 
second number 

(B) the first number plus the opposite of the second number 

(C) the first number minus the second number 

(D) the first nunnber plus the second number 



Key for Test F [1-92] 



1. 


A 


2. 


C 


3. 


A 


4. 


A 


5. 


B 


6. 


A 


7. 


B 


8. 


D 


9. 


C 


10. 


C 


11. 


C 


12. 


A 


13. 


A 


14. 


D 


15. 


B 


16. 


C 


17. 


A 


18. 


A 


19. 


C 


20. 


D 







Test G [2-22] 

I. No more than one answer to each problem is a true sentence. Which 
is it, if any ? 



1. (A) |3 - 10| = 3 - 10 
(C) |3 - 10| = I 10 - 3| 



(B) |3 - 10| = -| 10 - 3| 
(D) none of these 



2. (A) 3 + -3 < 3 - "3 
(C) 2 - 3 > 3 - 2 



(B) 3 + "3 / -(3 + -3) 
(D) none of these 



3. (A) < 

(C) (2 - 3) + -(3 - 2) = 



(B) 3 - " 10 = 3 + -10 
(D) none of these 



4. (A) '3x - 3' is a numeral 

(C) 'x* = 0' is an open 
sentence 



(B) x+y=y+x 
(D) none of these 



5. (A) a + a = 2a 
(C) c- 1 = c 



(B) b + = b 

(D) none of these 



- 17- 



II. Which substitution converts the open sentence into a true one? 

6. 3x = X + 6 

(A) '3' for 'x' (B) *0' for 'x' (C) 'y -6' for 'x' (D) none of these 

7. d'O = 

(A) any numeral for *d' (B) *x' for 'd' 

(C) 'x + -x' for 'd' (D) none of these 

8. X + y = 

(A) *-x' for 'y' (B) '3' for 'x' and '-3' for 'y' 

(C) '-y' for 'x* (D) none of these 

9. 2x - 5y = 34 

(A) '2' for 'x' and '-6" for 'y' (B) '22' for 'x' and '-2' for 'y' 
(C) '-2' for 'x' and '6' for 'y' (D) none of these 

10. xy = yx 

(A) any numeral for 'x* (B) 'x' for 'y' 

(C) any numeral for 'y' ^^) none of these 

\, 

III. Which substitution converts the open sentence into a false one? 

11. XX - 5x + 4 = 

(A) '4' for 'x' • (B) '1' for 'x' 

(C) '0' for 'x' (D) none of these 

12. (x - 2)(x + 3)(x - 5) = 

(A) '2' for 'x' (B) '-3' for 'x' 

(C) '5' for 'x' (D) none of these 

IV. Fronn which pattern-expression caji the given expression be obtained 
by substitution? 

13. 6 + 3x 

(A) 6 +XX (B) X + y (C) 2v + vx (D) (u + v)x 

14. 3x + 5x 

(A) ax + bx (B) XX + yx (C) (y + z)x (D) ^x + xy 



<^' 



':' Vo: 



:■* '<(>} 



bi\F. ■■■ 



rt'-'i (Ct 



;0.;^j.';'':gy i::--- ;,ri: 



■%-w - ^* - 



<' "'=": HV^ 



-'■ . ; I ^ 



s i;.. 



- 18 



15. 3a(b - 2c) 

(A) xa(y - xc) (B) 3x(x - y) (C) xy - z 



16. 3(x - l)(x - 4) 

(A) x{x - l)(x - 4) 
(C) b(x - a)(x - b) 

17. 9 +3(x - y) 
(A) X + 3y 

(C) X +y(x - y) 

18. 3a + 6b +c 

(A) uv + wx + y 
(C) {3x +6)(y +z) 

19. (2a + 10b)(cb) 
(A) (u + v)wx 
(C) (xa+5xy)z 

20. x(x + y) + y(x + y) 
(A) yx + zx 

(C ) xy + yy 

Key for Test G [2-22] 



(D) xy 



(B ) xyz 
(D) x(yz) 



(B) X + xy 

(D) z + X - y 

(B) X + (y +z) 

(D) xy + 2xz + c 



(B) (x +y)z 

(D) X + yz 



(B ) xy + yz 

(D) u(v + y) + v(v + y) 



1. 


C 


2. 


A 


3. 


A . 


4. 


C 


5. 


D 


6. 


A 


7. 


A 


8. 


B 


9. 


A 


10. 


D 


11. 


C 


12. 


D 


13. 


B 


14. 


A 


L5. 


D 


16. 


B 


17. 


A 


18. 


A 


19. 


B 


20. 


A 







Test H [2-38] 

I. Of which principle is the given generalization a consequence? 

1. For each a, 3 + a = a + 3. 

(A) apa (B) cpa (C) cpm 



2. For each b, (3 - b) + -(3 - b) = 

(A) ps (B) cpa (C) apa 



(D) none of these 



(D) none of these 



3. For each c, c = c * 1. 

(A) cpm (B) apnn (C) pml (D) none of these 



A. X- 



t! .>v>ar. 



ci (•'*• 



•'-• .'i>5 



19- 



4. For each d, d* 1 + d« 1 = d(l + 1) 

(A) idpma (B) dpma (C) apm (D) none of these 

5. For each e, lOe - 3e = lOe + -(3e). 

(A) po (B) ps (C) apa (D) none of these 

6. For each f, 3f + (7f + f ) = 3f + 7f + f. 

(A) idpma (B) cpa (C) apa (D) none of these 

7. For each g, (g - 3)g = (g - 3)(g + 0). 

(A) ps (B) dpma (C) pmO (D) none of these 

8. For each h, 3h + (3h + 4h)h = 3h + {4h + 3h)h. 

(A) dpnna (B) apa (C) cpa (D) none of these 

9. For each u, u{u' 1) +.0 = uu* 1+0. 

(A) apm (B) pml (C) paO (D) none of these 

10. For each x, for each y, (x + y)y * = (xy + yy) • 0. 

(A) dpma (B) apm (C) pmO (D) none of these 

II. Which ajiswer will make the completed generalization true? 

11. For each a, 2a + 3a = . 

(A) 5aa (B) 6aa (C) 5(a + a) (D) none of these 

12. For each x, x(x + 3) = . 

(A) XX + 3 (B) 2x + 3 (C) XX + 3x (D) none of these 

13. For each m, (3m - 10) - {3x - 10) = 



(A) (B) 6m - 20 (C) -6m +10 (D) none of these 

14. For each x, -^-x + jx = . 



5 2 7 

(A) ^x (B) ^x (C) jjX. (D) none of these 

15. For each w, (w + 2) + (w + 3) + (w + 4) = 



(A) www +9 (B) w + 9 (C) 3w + 9 (D) none of these 

16. For each k, (2k + 3)(k + 1) = 



(A) 5k + 1 (B) 3k+ 4 (C) 2kk +3 (D) none of these 



^'l 



:Hi;^J '••: 



i-L 4- I) 



C • (01 



ir'-5.r. 



.1 11& (ua; 



>>:i>(li.l-T ■■■ 



{••) 



i.'."''^ iv:; «', •'0.1 -"^ (■ 



•.»;-; :5r^ ..•> i<-'fi. ' i; 



■ ... t."f '» '.CSi ■ 



i.' .« : 



•V', ' 



• .t";-".": 



y.-A, 



•j^'' 



■■>-'; 



;!*i. ri}- 



itiiJ- tc, - ii? 



(a) 



;i!qr \ i-i; 



r -ria'.r' . 



»{*•. 



j>.:j •-..',' 



ii-^., 



•..■••3 -f. 

■':c:;: { 



'fT' 



C,:r^i.ii:?u- . :" :t:i^:i 



■ : ,:.; r;r .'i" Tr..- 






. iJ- 



! ) 



iy. I 



aii'iiii ^' 



:i I 



(;.•■■ 



•J-f 



•»: ■ .V 



c:-: ' n'-' »£;•.. ij 



li to) 



(v.y 



-r {.Al^ 



'H ■■> ■ :C; \(1) 



•M ■ •..♦, 



•v^ 1.7 



V ;'T; 






■it.i.nJ '•<■' ...>jn •{'il'> 



;i>iJ'; '{Ti, 



■■ -.'n 



;-- .-i»>:'. 






-20 



17. For each t, (2t)(3t)(5t) = 



(A) lot (B) lOttt (C) 30t (D) none of these 



18. For each u, 2(3u + 1)[5(4 + u)] = 



(A) 7(3u + l)(4 + u) (B) 10(4u+5) 

(C) 10(u + 4)(3u + 1) (D) none of these 

19. For each v, 3v + (v + 2)v = 



(A) 6v (B) vv + 5v (C) [3v + (v + 2)]v (D) none of these 



20. For each y, y(y + 1) + y(y + 2) = 



(A) y(2y + 3) (B) yy + 3y (C) 2yy + 3 (D) none of these 



Key for Test H [2-38]; 



1. 


B 


2. 


D 


3. 


C 


4. 


A 


5. 


B 


6. 


C 


7. 


D 


8. 


C 


9. 


A 


10. 


A 


11. 


D 


12. 


C 


13. 


A 


14. 


A 


15. 


C 


16. 


D 


17. 


D 


18. 


C 


19. 


B 


20. 


A 







'J^ 'i- 't- 



Recommended Reading 

Two books which we think would be of special interest and value to 
UICSM teachers are: 

Max Beberman An Emerging Program of Secondary School 
Mathematics 1958 

Jerome S. Bruner The Process of Education I960 

Both were published by the Harvard University Press, 

Two paperbacks from which all mathematics teachers should profit 
are: 

Irving Adler The New Mathematics Mentor Books, 1959 (50^) 

G. Polya How to Solve It: A New Aspect of Mathematical Method 
Second Edition. Doubleday Anchor Books, 1957 (95^) 



o, o^ o, 

'C "I" '1^ 



. riM 



,1 r><;-OJ' 



iK- 






;)■*:■■■ " •• ■no- ■•• 



r -r 



>•■,!.! i i-iiOJ' I "I i ■■ i.\-^ '^^ '•' 



J . • t 



•r ic un r (O 






;!.^r-: 



•?!••• ..'< 



.f.J 



Oj !ji!/ . 



^-.l » .'. 



.?!: ■». 



.1 .CJ 



! -lo't '^ 



^ 



.wTJit. 






.'l.j.; 



•, nil '• 



:■••(*. 



-21 



Quiz - Unit 4, page 4-41 

A. Suppose A = {-3, -Z, -1, 0, 1, 2, 3} 

B = {-5, -4, -3, -2, -1, 0, 1, 2. 3} 

1. How many ordered pairs are there in A X B? 



2. How many ordered pairs in A X B have first component -2? 



3. How many ordered pairs in A X B have second component greater 
than or equal to ? 

4. How many ordered pairs in A X B have first component less than —1 
and second component greater than 0? 

5. How nnany ordered pairs in A X B have first conaponent less than -1 
or second component greater than 0? 



B, On a picture of the number plane 
lattice indicate the dots which 
are pictures of ordered pairs of 
integers such that the first com- 
ponent is 1 more than the second 
component. 







....<. .<^ 




















-^ - - , , , . , 




^ -5 


. . . . ^ . . > 














^5 









I 



C. On a picture of the number 
plane lattice indicate the dots 
which are pictures of ordered 
pairs of integers such that the 
sum of the components of each 
ordered pair is 3. 



• * • • 



^^^^ 



-5 



• • • • » 



-> 



y 



u -• 



.&:■ J- 



. .. .i ,0 A 
,1 ,' A- .1-- ,^ 



;'..■(-. 



.1- - , f— f :- h 



:ic- 



.-r-irlJ s.\ 



-< A a: 



.■;;<.7 



t ».■■->*.■• a^ » - .. 



-22 



D. Use brace -notation to describe the sets pictured below 



(1) 



(2) 



-5 



■t 
5 • 

. \ 

HE 
BE 

4±t^ 



kJB 



I 



> 



E 



CD 



Q 



(i 



Q 



Q 



-i5 



E 



5 . , 



-> 



E. On a picture of the number plane graph the following: 



(1) y 



(2) X < 2 

















/ 
1 


i i i 




i 












MM 


i M 1 1 ! 












i ! 


M 1 i M 












1 ! 


; 1 1 ; i 




: ' ' ' U 


M M 




i , 1 ! i 5 


-4--.-- i 




i 


1 i ' J 


■ 1 


M ! 






i 


Mi ! 


i ! 


i 






MM' 








^ 


i ! j ' ' 






i : . 


x 






j 


X: : ^4- 


. 


i 






i 


T! ^rr^ 




i 




.1 1 


, • '■• 




^ i 1 




i ! 


' i i 


-- 






_ 1 LLJ__i_ 


! ' 1 1 

i ^ i ' 










i-^ 


i ; 1 






i 


I ! 


i ! 


i ■; 




^ ' !^ 




1 ' 












t 






\ 


M M 

















1 i _J ___ 


V 


! 




' 


1 








M i M 




1 








1 

i 








: 1 


1 




1 






J ( 


1 ; 




i 






1 ! 


1 i 




i 1 






! i 






1 






' i 


i 1 i 




1 I 

t i 




^ 


: 1 i ' 1 


i ' M 


^ 


•V 


-J5 I ; i i 


\\k\ 




j 


' 








1 i 




! 


; 


^ ■■ \ 






i 






1 


i ! 


: ; ! 


1 










! r 


! M : 


i 








1 
1 


—J 

1 




i 














1 


1 


i i 


i 














i 




i i ; 

i 1 




i 












1 


/ ' 















■lU-i-iXi 



••: ^ J 



23 - 



(3) y =x + 2 



(4) 



< 6 



















> 


\ 








1 














1 


1 
1 






i 


i ■! 


1 

1 










1 






1 






1 














1 




1 

1 


1 




^T r 














i 




ir- 


1 1 


n 1 r 






V 








P 


" ! 


! 1 ; ! 


1 1 




i I 








1 i : i 1 








i i 






; ! 


Mi 


" 






I 1 






; : 1 ' 
1 i ■ 1 i 


t^ 










1 : 

1 ! 


J 


1 i ' 1 ■ r ■ 
1 _ : 1 i 


V 








> 


1 1 




i-U-l^U 


1— ?* 








1 ' '' 




_4irTini:_ 










i i 


; 


^ 1 i M^ 




: 






i 1 




: 




M ' ' 




^-i_' 


! i_ 


«; 


', 1 


III i 




1 1 r 




! 1 r 


i ; 
; 1 


h ; 1 








i i ! 


' 1 
i 




i ! > 1 1 

1 1 ; i i 












i 1 






1 ; . 

l' i 1 














i i 
. 1 


1 


i 












M ^ i > 


' i 1 
/ 1 i 











- 




, . , : 


i 


1 


\ 


1 










1 i i 


i 


1 1 












1 


1 

1 


i i 












! 1 


1 
1 




j 






i 








! ! 

i 1 1 


|5 


1 

1 


1 






1 






._ 1 ll± 




1 


i 1 


i i 




1 1 


1 


i 


1 1 








' ! 

J_ ' 


j 


i i i 1 






r- I ■ 


I 


; 1 > 






~ — 


i h"! 


i 


1 ! iJ 






...»» 








1 \ "^ 








1 i ! i 






i 


1 


I 


1 1 h 






1 "1 

! 


i 




' 


- 


r 

i 


45 


^^ i 




! 


i 


— 1 |-1 1 






i 1 ! ! 


j 


i 


! 
1 








1 1 


1 1 




1 
1 


1 
















i ! i 






._ 


■ 


i . 1 


i 1 '■ 


1 : i j 



Graph each of the two equations and give the ordered pairs which are in 
the intersection of their solution sets. 



1. (a) X + y = 3 
(b) 2x + y = 6 



2. (a) |x - 2 I = y 
(b) |y| =2 













1 1 




/ 


\ , 










1 














I'M 


1 1 

1 


1 




















1 ' 1 






' 














M ! i 


1 






— 


— 












1 


1 


1 ! 










































1 






















j 


1 


1 
















-^ 






1 






i 
1 














■^1— 


— 






J 




i 


























_( 










i 

1 




c 








—p^ 




















i 








r 


























— 


\-] 






















1 
















~ 


1 












1 V- 


s 








■ 






















1 




■ 
















1 










i 




















— 








































! 






















i 
1 


1 


























\ 


1 ' 








1 

















1 




I 


A 




i 






1 

1 


! 
1 








! 1 


I 










— t— 










! 


1 




I 




1 

1 
















t 


1 1 i 


















1 


i 


( 












1 






_Trrl- 


1 


















1 
















1 i ■ i 






1 










Ml! ' 






"^ _i 


} 








1 i 1 5 1 


; 


^ 








i 




MM 






! 






i 




\ ^ \\ 


i 












1 ' i 

! 1 ! 


j 1'- 

1 








J 


, 


S 




1 
j 


















1 1 

, 1 1 






















1 




























t 












































1 


i ! 




1 










1 






, ! 1 






— i — 
i 






i 


i 


1 


I 


1 


i 




! 








! 


i- 





•(. L 



.c-:'0 '.'.n 



■•A= 



•■. ; :.■:•: 



; ( : -I -^ . 



; .1 



. ■ I 



-24 



G. Graph: 

1. {x, y): X < 2} 2. {(x, y): ly| > 3} 

3. {(x, y): X < 2 and |y| > 3} 4. {(x, y): x < 2 or |y| > 3} 

(2) 



(1) 






1 


/ 


1 




1 




1 






1 




IT ; i 


1 i 






1 




j 1 


1 










! . 




1 i 


1 


i i 








1 




-*- r-^--T 1 1 








! I 




4-4-t -r^-r 






j 




1 1 


1 


1 




i 






1 

L ; -Li_J 






j ; 






: 1 




1 

1 -N 


•'T' 






C I 




1 
1 


5 




j'' 






3 1 

1 






1 j 










1 

i 






1 i 1 








1 i 






III 

1 1, ' i 1 




t 
t 


; ' r u 


-^T-| I 




i 


! ri 


i i 1 ! , 




i 


! i 






II' 

i i ; ' 




1 ! 1 1 






! 1 i ! i 1 




_^ 1 n !_ 


, ! i 1 i 1 ! 




! 


1 1 ' 1 


1 i 1 1 ! i 


- 


i 
1 


M j i 


1 j 1 ! i i 








\ 


/ 


' 


1 




• ! 









A 












1 . 
ill 


1 


1 










M ' 1 


. ; 




i 








' i ' ' 


._^C_ - 


- ' i 


i 








I 1 ^ 

t ; 1 : 


! 1 




j 




1-- --I- J. 1 

■ i ! 






i 


1 
1 






' 1 i 








1 






_.._L,. ., 1 , 

: > ' 


1 


. .J 




- 




i 




1 


^^ ! t 


V 


* 


- ! 1 




! 




5 i ' 


■^ 




3 1 1 


' 


I 




i 1 




! r i ' ' 

' 1 1 ' 1 ' ' 


1 




I 
I 1 i 




i 


; 1 : . 








1 i 1 




1 


! 


f1- 


i i 


■ 

! 


1 




■ i I 


: 


i 1 


1 i ' ' 


! 1 


1,1: 


1 


1 I 1 ! I 


1 


< i 


i 


i 


1 1 1 


! ! 

i ' 1 


i ! 1 1 


! 


1 i ! 1 
1,11 




' i^_ 


i j i i 




111! 






: i 




\ 


/ 






■ 





(3) 










1 ■ h 


\ 










\ 1 








i 


i 




; 




1 






i 


1 
1 


i 


^ i i 


' 








1 


1 


! 1 
; 1 










i - 


i i : ■ i . 








L_ [!_?__ 


1 1 !■■■' 
' ' 1 






^ 7 i 


III 1 1 


"[" 








• ' i . 


i ■ 


i 
— i-_. 

1 




1 


i ; 


"T: i 


\ \ 




t^ 


1 


1 




r^ i-- 




■^ 


■^ 


1 


\ i i 






k 


1 ■ 








3 ■ 1 
, 1 1 ■ 




1 t • 


i 1 








' 


1 

1 ; 


1 


i 


1 


- 








i 


1 ; 


1 


i 


1 








i 


1 


— 


' ' 


1 








i 

1 


i— 3 - 
1 


: 1 ' 

! 1 ' 


i 










i ! 




i I ! 


i 












1 [■" 

1 ! 


^ \ 


! 








1 


i ; 


1 1 




1 1 


i 


1 




1 




1 1 , 


L j 














■ \ 


/ 








i ; 





/ 










: : , ' 


\ 
















i i ill ! 






"TM i i . 




1 "^ 1 T 




i_ Ul 


1 1 




1 

i 1 ' 






r 1 
1 1 


i 




1 i ; 






1 


i. 


j 


1 ■ 1 5 ~ 
II': 






' 1 


1 


i 




1 I ! 


' 






I 1 








! i 




1 
1 








"" 1 
1 










1 ! I 




1 


■ 


1 






^ 


; ! 
,^ i 


i 


[ i 




1 


i 






■-^ 


i 4 




^^A-i 


l-^ 




i : 


1 1 
1 


_r^ii 1^ 


\ 












i i 1 




i ! ' 










i 




j ' i 




! 1 1 










i 




^^rr 


i i 




1 




i -^ 


j 




i 
1 


1 
1 


1 i 1 




! i 


. :__.! ^_ 


! 1 

t 1 ' 






i ! i i 1 1 ! 


1 i ! 






1 1 ! 1 ' ' ' 
, 1 , L ! 1 ^ 


M ! ; ' ! i 












1 i ; 
\ 


/ 

















; ••( ; : ( • .:•:.• : 






(S 



••i) 



X'-l 



- 25 



NEWS AND NOTICES 

Mr. William Annett of Seaford High School, Seaford, L. I., New York, 
a new UICSM teacher and a member of the NETRC Film Study experimental 
group, will speak on First Course at the annual convention of the New York 
State Mathenaatics Teachers Association. 

Mr. Donald E. Palzere of the Edwin O. Smith School at the University 
of Connecticut, Storrs, was a panelist at the State Teacher's Convention 
in New Haven on October 28. He spoke as the UICSM representative in a 
presentation of new programs in school mathematics. 

Mrs. Mary Huzzard of Jenkintown, Pennsylvania, participated in a 
panel at Villanova University on October 28 in connection with Mr. Beber- 
tnan's appearance there. Mrs. Huzzard teaches at both Cheltenham Senior 
High School, Wynccte, and Ogontz Junior High School, Elkins Park. 

Mr. Beberman will speak at a meeting of the Illinois Association of 
School Administrators in Chicago on November 21. He subsequently 
will visit the following schools in addition to those listed in the last 
Newsletter: 

November 30 -- Permian High School, Odessa, Texas 

December 1 & 2 -- Tucson, Arizona, public schools 

5 -- San Diego, California, public schools 

6 -- Desert Sun School, Idyll wild, California 

7 -- E. M. Cope Junior High School, Redlands, Calif. 



REGIONAL ORIENTATION CONFERENCES IN MATHEMATICS 

A series of regional conferences designed to acquaint administrators 
and supervisors with curriculum revision projects such as UICSM has 
been taking place across the nation this autumn under the sponsorship of 
the National Council. The format of each meeting includes a panel of 
teachers who comment on their experiences in teaching one of the new 
courses, ROCM Regional Directors have supplied the names of the 
teacher -panelists representing UICSM at several of these conferences. 



Dates 
October 3-4 

October 10-11 



Location 
Philadelphia, Pa. 

Iowa City, Iowa 



UICSM Representative 

Sister Mary of the Angels 
St. Rosalia High School 
Pittsburgh, Pennsylvania 

Miss Grace Wandke 
Barrington High School 
Barrington, Illinois 



i-,»iJo, 



- 26 - 



Dates 
October 27-28 

November 3-4 



Location 
Atlanta, Georgia 

Portland, Oregon 



November 18-19 Los Angeles, Calif. 



Decennber 9-10 



Miami, Florida 



December 15-16 Cincinnati, Ohio 



UICSM Representative 

Miss Emma Mae Large 
Mary Holmes Junior College 
West Point, Mississippi 

Miss Dana Small 
Franklin High School 
Portland, Oregon 

Stewart Moredock 
Sacrannento State College 
Sacramento, California 

Robert Kansky 
Melbourne High School 
Melbourne, Florida 

Eugene Epperson 
Talawanda High School 
Oxford, Ohio 



Sonae Statistics 

Miss Eleanor McCoy, Associate Teacher Coordinator for the Project, 
reports the following data on cooperating schools for the 1960-61 school 
year: there are 144 such schools, located in 94 cities and in 28 states. 
274 teachers teach 521 UICSM classes with a total enrollment of 12, 186 
students. A breakdown by year of study is given below. 



Year of study in 
UICSM mathematics 

1st 

2nd 

3rd 

4th 

5th 



No. of 



asses 


Enrollment 


299 


7109 


142 


3510 


66 


1324 


12 


226 


2 


17 



■:oiy 



mnf' ' :ri 





ews et"(2r 



An occasional publication of the 
UNIVERSITY OF ILLINOIS COiMMITTEE ON SCHOOL MATHEMATICS 

1208 West Springfield 
Urbana, Illinois 



'Simplify' 2 

Another Use for the Principal 

Operator 5 

Proving the Division Theorem 8 

■/S : Rational or Irrational? 12 

A Teacher's View of UICSM 20 

News and Notices 24 



UICSM Newsletter No. 3 



January 9, 1961 



H- -r- 



' ' V C V V >x3' ' ••• ' • • 

1 V...- -^.^ I V-/ • * ^»— - v <■' 



aDlTAiviaHT.-'M ., 



■ MO . on ^lOi^! .1,11 ?0 YTI' 



n*: 






MX:;I'7 to v/-.'^ a*:! 



I »S BW914 



;.aoajBl ., , 



.oi/I ■X9tj»iew9;/1 



2 - 



'SIMPLIFY' 



The word 'simplify' is used in so many ways that it is impossible to decide 
that any particular expression is simpler than another equivalent expression. 
Whenever you read 'sinnpler' ask: "Simpler for what purpose?" 

(1) How many pennies are in the box? If the number of 
pennies is fifty-seven, we would all feel that *57* is 
a better name than '3»19'. 

But consider the question: If nineteen pencils cost 
fifty- seven cents, how much does each pencil cost? 
If we use the name '3 •19' in our thoughts it is much 
easier to realize immediately that each pencil costs 
3 cents. This is a case in which '3» 19' is a simpler 
name for 57 than '57' is. 

(2) Which of these two equivalent expressions is the simpler? 

2(19x'- 7) 

38x - 14 

If we are concerned with the number of operations involved 
after a numeral replaces 'x', we see that the expression: 

2. (19- 3-7) 

involves 19*3, 57 - 7, and 2-50; we have three operations 
here. On the other hand, the expression: 

38-3 - 14 

involves 38*3 and 114 - 14--only two operations. 

If we replace 'x' by '3' and evaluate expressions such as: 

n\ 38x - 14 

^*' 19x - 7 

and: 

(2) 2(19x- 7) 

^^' 19x - 7 

we find that (1) involves five operations while (2) can be 
simplified quickly by using: 

V V / ^= X 

^xV/0 y "" 

So, for this purpose, we would consider '2(19x - 7)* simpler 
than '38x - 14'. 

^e should begin (as soon as possible) to teach the child that the meaning of 
[the word 'simplify' must be considered relative to the problem in which 



- 3 



we are engaged. Try to do this by stressing the principal operation that is 
involved in an expression. The first tinne (and nnany other:times, for a 
period of several days or even weeks) that we consider the theorem: 

V V / V / ^^= - 
X y / z / yz y 

point out that in order to use this generalization we must have multiplication 
as the principal operation for both numerator and denominator. 

Long before this, a discussion similar to the following should have occurred. 

Consider the pattern sentence: 

(1) a • (b + c ) = a -b + a -c 

What is the principal operation indicated in the expression which is 
underlined once ? What is the principal operation in the expression 
which is underlined twice? So (1) gives us a pattern which we can 
use to find an expression in wh|ch the principal operation is addition 
and which is equivalent to a given expression in which the principal 
operation in multiplication. Sentence (1) also gives us a pattern: 

a'b+ a«c = a«(b+ c) 

which we can use to find an expression in which the principal operation 
is multiplication and which is equivalent to a given expression in which 
the principal operation is addition. 

In some cases it is more convenient that the principal operation be 
multiplication. In others, we can do our work more rapidly if the 
principal operation is addition. Much of the practice that we are 
doing now will prepare us for more complicated problenas later. 
When that tinne comes, we must decide whether it will be simpler 
to use an expression such as: 

a«(b + c) 

or to use an expression such as: 

ab + ac 

P We can only decide this after we have encountered the problem. At 
that time we must be able to change fronn one expression to an equiva- 
lent one rapidly. 

To go back to: 

WW W X. * z _ x 

^x y ^ ^z ;^ yz y 

If we try to 'reduce a fraction' we must first be certain that the princi- 
pal operation in both numerator and denominator is multiplication. You 
may find that this is a very effective way to help students avoid the 
following error: 

ac + be , , 

= ac + b 

c 

Of course, this work on principal operation starts back in Unit 1 when we 
are finding unstructured names for such nunnbers as 3 + 8»7 and (3 + 8) '7, 



■■I n 



■•.A'. 



'■■' .-'.'- 



- 4 



When we get to an exercise such as: 

3 2 

Simplify: 7^ ■*" Ty 

we point out that for some purposes we consider an expression in which 
the principal operation is division simpler than one in which the principal 
operation is addition. So, such exercises are designed to give us practice 
in making such changes quickly and correctly. 

Some of the difficulty arising from the use of the word 'simplify' is due 
to the child's earlier training. We are all inclined to feel that we have 
not multiplied 3 by 19 unless we write (or say) *57'. When we see '3 •19', 
we construe this as a command to find the equivalent standard decimal 
numeral (which is '5?'). As arithmetic teachers we say 

What is 3 times 19? 

and we expect '57' for the answer to the question. We mean: 

What is the standard decimal nuineral for the product of 3 by 19? 

We must begin our study of the word 'sinnplify' as soon as possible. If 
we can begin in- elementary school, so much the better. If we have to 
start in the higher grades, let's start then. --A.H, 



VI- 



- 5 



ANOTHER USE FOR PRINCIPAL OPERATOR 

The use of the principal operator (Newsletter 1) to determine if a certain 
sentence is an instance of a given principle is actually a rather compli- 
cated form of another use of the principal operator. To illustrate: 

In order to show that: 

^^,j 7 + (3 + 2) = (7 + 3) + 2 

is an instance of the associative principle for addition, we examine 
the pattern sentence: 

(**) ( + ) + = + ( .. + ). 

We must decide whether or not either side of (*) will "fit" the first 
side of (*-). Having decided that the second side of (*) "fits" the 
first side of (**), we must then decide whether or not (with the same 
substitutions) the first side of (*) "fits" the second side of (**). In 
effect, our first step is to examine the two pattern express ions ♦ 

(a) ( + ___) + (b) + ( .. + ) 

and then to determine if either '7 + (3 + 2)' or '(7 + 3) + 2' can be 
generated from (a) or (b). Having decided that '7 + (3 + 2)' can be 

generated from (b) by substituting '7' for ' ', '3' for ' ', and 

'2' for '^^VU', we must then decide whether or not '(7 + 3) + 2' can 
be generated from (a) by these same substitutions. 

Further practice of this kind occurs in Unit 2 on page 2-21. These 
exercises have often proved difficult for students (and sometimes 
for teachers). 

Let's examine the first of the twenty-four expressions at the bottom of 
the page. We must make fifteen decisions about '9+5 '6'. [So, in doing 
this work the student must make 24*15 (that is, 360) decisions. Thank 
heavens, many of these are very simple.] So, here we are with: 

9+5-6 

The principal operator is '+'. So, '9 + 5-6 can only fit a pattern in which 
the principal operator is '+' or one in which there is no principal operator. 



:/• 



:^rt:cnt."-» 






■jj," ill- (•■■'l ':■ 



"(^11' ■;..";.''- '-V--' 



;ia> '\ ' <» 



-« I 



^'--^'.iVf 



- 6 - 



[A single pronumeral is a pattern expression without a principal operator. 
In this exercise, we are given no patterns consisting of a single pro- 
numeral.] We innmediately discard patterns (b), (f), (h), (j), (m), (n), 
and (o). Now let's examine the ones that remain. 



(a) x+ y 



9+5-6 
(c) y + X 

A^LZ 



Yes. Substitute *9' for 'x' and '5 -6' for 'y'- It 

may help to use ' ' and ' ' or frames at the 

beginning. 

Yes. Substitute '9' for 'y' and '5-6' for 'x'. 



9+5-6 



(d) xy + z 

~ 'fmj + 

9+5-6 



(e) a + be 
9+ 5'6 



(g) ab + ac 
9+5-6 



No. Remennber that we must substitute for each 
pronumeral in the pattern. We can substitute 
'5 -6' for 'z', but without changing '9' to an equiva- 
lent expression (which is not permissible in this 
exercise), we can find no substitutions for 'x' and 

y • 

Yes, Substitute \9' for *a', '5' for 'b', and '6* 
for 'c'. Notice that after matching the principal 
operators we have two new problems. Can you 
generate the expression '9' from the expression 
'a'? Can you generate '5 • 6' from 'be' ? 

No. See (d) above. 



(i) uw + vw 
9+5-6 



No. See (d) above. 



(k) X + (y + z) 
9+5-6 



( ) ' a + b I + c 



No. We can generate '9' from 'x'. We cannot 
generate '5 • 6' from 'y + z' because the principal 
operators are not the same. 

No. See (d) above. 



9+5-6 



r>Ci:. 



' kW\ 



.Jr. 



■•'^I'^. 



..4i..<>''i^ 



.W-' 



- 7 - 



Now let's see from which of the fifteen pattern expressions the thirteenth 
expression at the bottom of page 2-21 can be generated. First, take a 
good look at the expression: 

(2a + 3b)5c. 

If any pattern expression is going to generate this expression, we naust 
be able to "nnatch" the principal operators. Let's unabbreviate. 

{2a + 2b)5c = (2a + 3b) • 5 • c 

= [(2a + 3b).5]-c 

So the principal operator is the second ' • '. It follows that this expression 
can only be generated from a pattern expression whose principal operator 



IS • 



Let's look at them. 



(b) ab ,_, 

A- D 

[(2a + 3b)- 5 -c 



(f) x.(y+ z) 

[(2a + 3b).5].c 

(b) (m + n) ■ p 

[(2a + 3b) .5]- c 



Yes. Substitute 'c' for 'b' and 
'[(2a + 3b)- 5]' for 'a*. 

No. We cannot find substitutions 
for 'y ' and 'z '. 

No,. We can substitute 'c' for 'p'. 
However, when we try to generate 
'(2a + 3b) • 5' from 'm + n' we find 
that the principal operators do not 
match. 



(j) P-(Q-R) 

[(2a + 3b) -5] • c 

(o) [x .y] -z 

[(2a + 5b) '5] -c 



No. We cannot find substitutions 
for 'Q' and 'R'. 

Yes. Substitute 'c' for *z', '5' for 
'y', and '(2a + 5b)' for 'x'. 



The process by which one determines if a given expression can be 
generated from a certain pattern expression is purely mechanical. No 
principles for real numbers are involved A. H, 






'\ . bv J. •••;:.■••••• <'y 



v-.-.i'''!' 'jv*' ;•••' ' 
.-..1 ^ ■• ■■ 



.•»fJ! 



^1 



:!>>.• •!>.? :aX ' '-U''' ■' ■'■ 



jj.i •- 'i;:.': j;. .<(' . 



)l 



.!?•(:■• 



n 



'r.'- Vol ' :' -..'r 






t: 



• iif < 



•.7t/ijJ ■. ".'JI-l;"- ;>■< : 



• «•..:' . > 



■ <.■■■ 1. 



8 - 



PROVING THE DIVISION THEOREM 



Proving the division theorem, pages 2-89 and 2-90, in Unit 2 may be 
difficult for some students. Mr. Jones and Mr. Edwards of Zabranchburg 
Junior High School had some ideas about this development. We thought 
you might find it interesting. If you duplicate and use it, we would like to 
know how it goes. 



Mr. Edwards Teaches the 
Zabranchburg High Math Class Again ! 

Remember Mr. Jones who teaches mathematics in Zabranchburg Junior 
High School and the principal, Mr. Edwards? Mr. Edwards took the class 
when Mr. Jones was ill. Mr. Edwards had not gone to the same college that 
Mr. Jones had. However, he knew that the class was studying basic principles 
and proving theorems. He couldn't stay in the roona all hour but he wrote some 
homework, had his secretary duplicate it and told the students that they could 
have the hour to do the work. This'is a copy of the assignment sheet. 



Basic Principles 
(1 



(2 
(3 
(4 
(5 
(6 
(7 
(8 
(9 



V V, a+b=b+a 
a b 

V V, a • b = b • a 
a b 

VV^V a + b+c=a+(b+c) 
a b c 

V V, V abc - a • (be) 

a b c ^ 

V V^ V (a + b) -c = ac + be 
abc 

V a + = a 
a 

V a • 1 = a 
a 

V V^ (a - b) + b = a 
a b 

"^a^^O (a -M.b = a 



[cpa] 

[cpm] 

[apa] 

[apm] 

[dp ma] 

[paO] 

[pml] 

[pr] 

[pq] 



Assignment. Using only these nine basic principles, and theorems you 
prove using them, prove: 

VVV ifz+y = x, then z = x - y. 
X y z ^ ' 



•^^ vi^ v'^ 

'1^ 'r '!•• 



■Even Fred, "The Brain", was puzzled. He knew what to do if he could use 
the po and the ps. But Fred knew this wouldn't do because Mr. Edwards 
|said use only the basic principles he had given them and theorems they could 
Iprove using these principles. Suddenly Fred thought--just maybe he could 



.1 i if. 
{"■■'■ ■■ ■ 






■-•rf' p 



, l-=<iJ 



ij.><' v;i-.-.": 



..?::. IC. 



- 9 - 

Drove a theorem that would take the place of the po. He decided that 'pr' was 
in abbreviation for 'principle of remadnders' and that he would have to use 
±iat. 

lust what is the pr ? It is given this way: 

V V (a - b) + b = a 

Mow what does that mean? Fred decided to look at some instances of it. He 
;an get an instance by replacing 'a' by the name of any real number he decides 
:o choose. So he got: 

V^ (1 - b) +b = 1 
b 

Vj^ (8 - b) + b = 8 

V^ (-2 - b) + b = -2 
b 

^- V^ (0 - b) + b = 

b 

Suddenly Fred said, "Oh". That last instance looked mighty good. He decided 
le had found something he could use'instead of the po, because 

if V, (0 - b) + b = 

D 

then V^ b + (0 - b) = [cpa], 

}0 anytime he wanted to use '-b' he could use '0 - b' in place of '— b'. He 
iecided to call this theorem the 'ot' [Opposites Theorem]. 

rred thought things were going pretty well, but he still was a little worried. 
Vlaybe he had better see just what he could.do with the Opposites Theorem. 
iVhat would he try it on? Well, the cancellation theorem for addition was used 
I lot. Maybe he. could prove it using his new theorem. Just how would he prove: 

V V, V if a. + b = c + b, then a = c. 
a b c 

■^e thought about the way he would do it if he had the po. [Think about it]. 

Suppose that a + b = c + b. 

(a + b) + (0 - b) = (c + b) + (0 - b) [upa] 

a + [b + (0 - b)] = c + [b + (0 - b)] [apa] 

a + = c +0 [ot] 

a = c [paO] 

Hence, ifa +b =c +b, then a = c. 

red was pleased. He could use his ot instead of using the old po. He decided 
9 tackle the assignment. Prove: 

VVV ifz+y=x, then z = x - y. 
X y z ■' •' 



:ri I.: 






C-;T!/ 



■' •..•' 'n -.r:"' 



■.::■<.:•. 



o-'Vr . :,::n- H .t'.cv {Hijj'M'. I';;"-;^; 



■» -■ .» 



.• 'li- :>v'j- > iif-j-'l"'" 



:l'!^;-:v;"r'''v 



I- 



oi> :'.'i. 



tyir- 



!):'.' •.•). >jrjh;.' 






- iO 



•' /^T^>« 



OK", Fred thought, "here goes". 

Suppose that (1) z + y = x 

Fred decided he had to get a subtraction sign into the proof. The only- 
generalization that had a ' -' in it was the pr. 

(2) V^Vy (x - y)+ y= X 

He sat and stared at (1) and (2). Then he realized that from the two of 
them, he could substitute '(x - y) + y' for 'x' and get: 

z + y = (x - y) + y. 

Since he had already proved the cancellation theorem for addition, he 
could get: 

z = (x - y). 
So, if z + y = X then z ~ x — y. 

He decided that Mr. Edwards knew some mathematics after all. Just 
then the bell rang. 

Mr. Jones didn't have a chance to see Mr. Edwards before the next class 
meeting. When the class started, Mr. Jones told the students he wanted 
to introduce the new work on division first so that they could be sure to have 
enough time to set the stage for the next day's assignment. Then they'd 
have a discussion at the oi.d of the period, Mr. Jones wrote the Principle 
of Quotients on the board. , 

^a^b/0<^ -^^-^^^ 
Then he asked them for some instances of it and wrote them on the board. 

(2 -^ 3) . 3 ^ 2 

(j-S) = 8= i 

(0 ■:- 2) . 2 = 

(i ^ 7) • 7 = 1 

{-5 V 5) • 3 = -5 

(4 ^ -2) . ~2 = 4 

(-6 i- -7) • -7 = "6 



vfr'' •<'''' .i^M-yj ;;r'. 



ic. 



v. -•: .■■•.• '. ':< '-•» . ■<;;■: 
-•.'•" -P-'v*i.. 



.wCmOvi ■ i!? ;.o in-. 



,.t. ; 



11 - 



About this time Fred started digging for his scratch paper from the previous 
day. Where were his instances of Mr. Edwards' pr ? Here they were: 

V^ (1 - b) + b = 1 

V^ (8 - b) + b = 8 
b 

My, but these were like the ones on the board. He wondered if the succeeding 
proofs would also show a similarity. Just then Mr. Jones said, "Can anyone 
think of a generalization we might be able to prove immediately by using the 
pq? 

Fred had one. [What do you think it was ?] 

V^ ^ (1 ^ b) = b = 1 

Even Sammy thought that was pretty cool because it was just an instance of 
the pq. 

Mr. Jones asked for another generalization. By now Fred was sure that 
the cancellation theorem for multiplication could be proved using the pq, 

V^ V, / A V if ab = cb, then a = c 
a b 7 (J c 

[How would you prove it? Hint: Go back to Fred's proof of the cancellation 
theorem for addition. What will replace '0 - b'? '+'? '-'?] 

The assignment for the next day was to prove: 

V V V A V if z '-y = X, then z = x■^ y (= — ). 
xyfOz '^y' 

Fred decided this assignment wouldn't be very hard. --A. H. 



I 






h*<n 



(«» 



■,1-7 |. 



• •■••* J. 



4i .. 



' ■'> V^-. 



•'•i-.. 



'^••' {Ji-f.t 



■Ji.- 



■'str' . ♦ 



12 - 



•v/S": RATIONAL OR IRRATIONAL? 



Students may have some difficulty understanding the proof that V8 is irra- 
tional unless there has been sonne special preparation. Here is a suggestion 
for such a development. We'll use . 52 as an example. 

Teacher: You have said that . 52 is a rational number. Can you justify that 
statement ? 

Student: Yes. K there are two integers whose quotient is ,52 then .52 is 
rational. One name for .52 is 'tttjt' or '52 -r 100'. So there are 
two integers, 52 and 100, whose quotient is .52. 



52 



52, 

loO 



6Z r 100 



Teacher: Can you give me another name for .5 2, in which the principal 
operator is '-r' ? 



If. 26^50. 



Student: 

Teacher: Another one. Another one. Another one. 



.52 


- 


SZ ^ 


/ oo 


Z^ ^ 


50 


13 T 


25 


104^ -^ 


2,00 


^08 ^ 


4-00 


-5? -^ 


- loo 


-Zto T 


-SO 



Teacher: Let's throw out all negative divisors. Can you fill in the blanks 
so we will have more numerals for . 52? 



- 13 



5Z T loo 

2fo -r 50 

13 t ^5 

I 04-T too 

ZoSr 4-00 




Student: 39, 65, 260 
Teacher: How about 



4- 10. 



Student: Well, 5t -^ 10 is . 52, but 5 — is not an integer. 

Teacher: How many more numerals like these could we write? 

Student: Many. 

Teacher: Now really stretch your imaginations. Think of all the possible 

numerals for . 52 that are of this kind. Remember; throw out all 
negative divisors. Now think of all the divisors. 



.5 2- 


52. T 


/loo ~^-^ 


£(b -r 


50 \ 


>3 -^ 


ZS \ 


|o4- ^ 


^oo \ 


ZoQ -r 


4- 00 \ 


39 ^ 


75^~^ 


65 T 


IZS / 


2.GO -T 


500 J 


> 


• 



14 



Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 

Student: 
Teacher: 



How many numbers are there in this set? 

Lots and lots. 

What is the largest number in that set? 

500 

Stretch your imagination some more-- 

There isn't a largest. 

What is the smallest number in that set? 

25 

You mean that no one can find a name of this type for . 52 such 

that the principal operator is 'v' and the divisor is less than 25? 

That's right! 

Now, let's fill in these blanks. 



.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 


X 
X 
X 
X 
X 
X 
X 
X 


100 = 


50 = 


25 = 


200 = 


400 = 


75 = 


125 = 


500 = 





Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 



What kind of number did you get in each case? 
Real number. 

That's corr-^ct. What kind of real number? 
Positive, 

Right. What kind of positive real number? 
Rational. 

Right. What kind of positive rational number? 
Integer. 

[We hop3 it won't take this long to get this answer.] 
Do you think you can find a positive integer smaller than 25 such 
that if we nnultiply . 52 by that integer the product will be an inte- 
ger? How many positive integers are there that are snnaller than 
25? 



15 



Student: 24 

Teacher: Let's try them. Mary, you multiply .52 by 1. John, multiply 

.52 by 2, Kate, multiply .52 by 3. Fred, multiply .52 X 24. 

Did anyone get an integer for the product? 
Student: No. 
Teacher: Then, 



ZS IS 


THa SiMf\LL£5." 


T Pc^lT 


we. 


INT 


E6.ER. 




^ o<:^ K 


Trt AT 


T t^.tl 


PR.OPOC.T 


OF 


.5^ 


6Y 


THPkT 


|MT£(5 


BR. 1 


^ /\M 


ifoTe.<s>£R.<, 





Teacher: Do you thixil^ that v/e could do the sanne sort of thing for . 125? 

Student: Yes. 

Teacher: How about . 3? ^? 

for any rational number? 
Student: Yes. 



6? -- ? Do you believe we could do this 



When you are ready for page 4-48, you nnight do sonriething like: 

Teacher: Let's think about nTs^, The text tells us [bottom of page 4-47] 

that 'vS is not rational. Let's prove it. Suppose it were rational 
Now if it's ^-cally not rational and we suppose it is what is going 
to happen? 

Student: We'll be in a mess. 

Student: Everything will get fouled up. 

Teacher: Right. 



These answers were actually given 
in a class where this was tried. 



PR.C 


:j\j el 


-is 


IS 


MOT 


RATlOK^AL- 


'So 


PPo^ 


^ ^^8 


1^ 


R^-T 


lOKJ AV_ 






[Nov. quickl/ review the development that led to 

25 is tho onaall'^st positive integer such that the product of ( 
c 52 by that integer is an integer.] 

Teacher: Jf n/S is rational, can you inake a sentence like this one? 

Student: Yes, ^ 

Teacher: What naarks in this sentence will have to be replaced? 

Student: '25' and ,52". 



•?.!f 



- 16 



Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 



What shall W3 put in place of '.5 2'? 

'nTs'. 

Do you know what to put in place of '25'? 

No, 

Let's use a pronumeral, 'q'. 

Let's call thc.t product integer 'p'. 



Cj /t> TKE SfY,Al_u^-il 


Pos>T\V£- iNjT£&a.R 


<Sl> C- \A T H /N.T ^f^. c^^ 


1^ ^^J \Nie-0»E,FL 


^°- ?- = 




f 



Student: 



Teacher: 



Student: 
Teacher: 



How shall I fill in the blank? 

'nTs'-. 

[Now, you ar3 ready for line 5 on page 4-48.] 

I am thinking of a number that is a positive integer and it is less 

than q. Let'c call it 'r'. What can you tell me about r? Let's 

call it 'r'. 

It's rational. 

That's true. What el?e? [If necessary, say]. Look at this 

sentence on the board. 



■^ IS TME 
iNTl£.Gn£.R S^ucH 


thkt 


ST 


Posawe 


As^s) 


INT 


'b-O^E-R-. 











What can yoo tell me about ^/8 • r? 
Student: nTs"' r is not an integer. 
Teacher: Right! If yov multiply n/8 by a positive integer which is less than 

q the product is not an integer. Let's put that up here. 



'f IS T\Aa •=~,^Af^L-UE.'bT POS\TiVE Hs!TE6.£R 

-vTB -(a^'Y Po5.it we INT£6,£.R. LCbS Ttl ^Kj <^) lb WOT /^tO IMTE^ipR 



Now, let's think about n/s. Can you tell me an integer that is 

larger than N 8? 
Student: 10, 9, 1,000.000, 3. 
Teacher: One that i.s sinaller than "^rs"? 



.er. 



hI fl . N.- '^•'" '-N" 



17 



Student: 
Teacher: 



Student: 
Teachert 

Student: 
Teacher: 

Student: 

Teacher: 

Student: 

Teacher: 



Student: 
Teacher: 



-10, -1, UOO, OOO, 0, i, 2. 
Let's use 2 and 3, So 

2 v: ^^s" < 3 
Or if we use ' £' instead of ''^' we write' 2 < ^ < 3'. 

q q 

Now let's use the transformation principles and get some equi- 
valent inequations, 

2q < p < 3c [t > cizid the multiplication principle for inequations.] 
Can you get anotl cr ens that begins 

< 
< S - 2q) < q 

So what do v/e knovy about p - 2q? Can it be 0? Can it be negative? 
Can it be as large cs q? 
(p - 2q) is positive rrd'ir less than q. 
What kind of nvirabur is (p - 2q)? 

Integer, The set of integers is closed under multiplication and 
suhti action. 

So (p - 2q) is a p03itive integer that is less than q. Now finish 
this sentence: 

'■T^' (p - 2q) is 

Not an in'-eger i 

Let's ucc ' -' i-ar-L-ad of '-/S'. 



/^ 



-^■[F-Zf) 1^ ^C>T ^^o |N)-TE.G,£R- 



Student: 



Lets v.T'te ,his -n some other form. What expression is equi- 
vale lit to ' ■ '-^ - ?ri)''. 



P' 



(-^'-2.f^) I^MOT AM \t^TE<^B.R. 



■in^ty' 



use- 



•rn ■.'?''j--Nit'.;f-!" v -■-iJ. : 



• ,iV -' ■ .:.i<-.» ' 



• . o fi •■> i ' -■' •■* : * ■' ^' ' - -'^ ' '-*■' ■ •■•■''^'* '■ ^' 



-.tr.-i.i.' - 



•:i- ■■.r. "J.n 



•i-i iV.v;/-t-v> . ;.'• 



r.Mi-' ;-o'- ••.'.•i?':i*A''''' " T-'W"'* i>--- ^■'i 



^-*• .-'i.' 



'tvf'^J- V";../i .[■' •'I'^O-' ^.- 



>i ^e 5-.;(i: 



iijiv-v •♦'i noj".*-^r;'.'.%i..<o .-•. 



'. ■ c*i 



- 18 



Teacher: Let's get really clever now and find another expression that is 
equivalent to 

q 

What is one principle that is very useful in getting an equivalent 

fraction? 
Student: Principle of mulitplying by 1. 
Teacher: Now think of expressions that are equivalent to 1 and we'll try 

some of them in the blank. 

- 2p is not an integer 



p; 

q' 



Student: 






T> ' 2 q 

Teacher: Let's use that last one. 



-P- .a 
q q 



2p) is not an integer 



n n 

Student: So [(^-^ - 2p)] is hot an integer 



Teacher: I rather like that. Look at this part of it. 




Z- 



2,^) |S MOT AM 1 N T E^ cs, £;:?_ 



Student: 
Teacher: 



Student: 
Teacher: 



How else could this be written? 

[So ( — ) q - 2p] is not an integer 

P 
Say, just what was — anyway? 

^/8" 
So, 



^aTs")^- % -2.p] IS MOT AN |NiTE6,bR. 



Student: But, but but (^/^)^ is 8 and 8]_s an integer. So 8q is an 

integer and 2q is an integer. So, (8q -.'2) is an integer, 
^^aciiert How about that ? What's happened? 
Student: We've made a mistake? 
Teacher: Let's see if we did. What was our original problem? 



i. i J >:>{{ ! 



'ntvtA'a »■' ' 



•f;:>l >;'-i . :ij5 ^niJioi^.::: jj. 



■II,! .■■.»f."-^- U. ■«^-C'i.»i'. 



■ 1 ' 






■' ■•'■•;>■• .'I.. ■> . 



iV.J- 



':»*»'" ■•'■•>'.'? 



.^...^>•^.^ 



\:? 



..h .'tin- 



Vfiv/; \ 



•\'t't;.--l 



: c- .-> 



r^ivrt- 









.'' " r' i 



-«lil •?? 



y.Oj; 



/n -iqu 



r!>.'.Jl 



- 19 



Student: Show v8 is not rational 

Teacher: How were we going to do that? 

Student: By supposing nTS were rational and that that would foul things up. 

Teacher: Well, has that supposition "fouled things up"? 

Student: Yes. 

Teacher: Let's stop calling this a "mess" and say that: 

"Our supposition has led to two contradictory statements". 

Now what do you conclude ? 
Student: That n/s is not rational. 



Show vS is not rational 

Suppose n/S is rational 

q is the smallest positive integer such that n/S • q is an integer (p). 
So - = -/S 

q 

'v/8^' (any positive integer less than q) is not an integer 
2 <48'< 3 



2 < 


- < 

q 


3 


2q< p 


< 3q 


< 


(P ■ 


- 2q) 



So (p - 2q) is a positive integer less than q. 
Hence */^(v - 2:^) is not an integer 



— • (p - 2q) is not an integer 
I— - 2pi IS not an integer 



r q , 



2p| is not an integer 



rP^q 



- 2pl is not an integer 



[ (— )^ • q - 2p] is not an integer 

q 



[(n/S)^' q - 2p] is not an integer 



{8q - 2p) is not an integer 
but it is an intege 

So "/s" is not rational. --A. H, 



Contradiction 



.j.:b::.iuo' ^•»'':('i#^ '' 



. -ir-i-ji^i' r-^ 



.La; '■ • M J-^^IV • ^? 



'•;>'■ 



^.m 



! I ! i 






l> 



) 



" zo 



A TEACHER S VIEY/ OF UICSM 



(This article is a slightly edited verr-i'-n of the r.psech given by Mr. Howard 
Marston of the Prir.ci-^ia Upper School Et, Louis, Missouri, as his contri- 
bution to the panel of teachers who coiMinented upon their experiences with 
i^e several new school rr.athemat:cf> prograrno at the Topeka, Kansas, Regional 
Orientation Conference ci administrator.- and supervisors.) 

^ I think it should be made clear tha.t these various programs in de- 

veloping mathenn^tics curricula are nc ■!: in conapetition with each other. Each 
is organized in a diffcre-t vay, snc. £?ch serves a somewhat different purpose. 

We are presently in cur sixth year of teaching the UICSM materials at 
Principia in St. Louis. The UICSM pi L\frvarn had its beginnings in 1951, and 
in 1955 we entered the prograna as the fifth school to join as a "pilot" school. 
There are now 144 cooperating pchoolt.. 

V/e had heard a talk by one of tie staff telling what they were doing at 
Illinois; we wrote thorn letters and tal <.el with them until we were finally 
accepted as one of tb i 3chools in v/hicii to try their newly written materials. 

fOur mathematics department wanted ti try out the materials, and so did the 
administration. (It is in':eresting to n* ts that both our headmaster and 
assistant headmastvjr were former ina-.L teachers. ) We were all of the opinion 
that the students couldn't learn any le.TS m?thenaa-ticE than they were learning 
in the traditiona.1 classes, and perhaps 'hey would learn more. 

We began that ia.ri of '55 by ha.\ing brother teacher and myself each 
teach one 9th grade UICSM class. We had had no •special training for teach- 
ing these materials. I n^yself had tau'jht co.ivcntional mathematics at public 
and private schccls in tha East for five years ^prior to this, and the other 
.teacher had taught mathematics for m '.^iv yc:irL- 

I • Not only did we not h?,ve spccic.l ti'^ining at *he time, but we did not 

even have all the matsriaic. The m?.t';i'iali;: in mimeographed form, were 

.being mailed to us almost on a day-to-day basis a.t times, and usually the 
commentary for teachers arrived too j-'.tc. M'^;: Beberman and David Page 
rpade several visits to our school duvi.-K the year and we visited them in 
Urbana three or four tinges to receive Lho loA-'-v-t information and instruction. 
Of course, we kept in contact by mail .^.nd soinetiines in emergencies by 
telephone. It was net long before v/e v ere coi~ipletely sold on the program 
despite those early inconveniences., rr d ^o v/ere our students. 

What we l.ked inost wa.s the Ii-c': thc.t th3 ctudents had to think and, 
although this was a new experience foy thorn, thsy began to enjoy it. Oh,, 
:here were one or two stv.dents who sa." ''.Tvj.t give me the rules and I'll do 
:he work. " But r-.ott students scon er . ;rf d into the spirit of the game. We 
discourage students from helping cr.f^. .Hncler {we like to let each student dis- 
cover his own rules and short cute taiou-^h careful development of the exer- 
;ises), and we disco iragB the rtudent: f.'.-om. ^stting help from their parents, 
rt isn't long before the parents rre *'lc'it''' a:iyv/ay„ 

' This brings up the point c^ parent •■.<!' o*icn to the program. Every so 

)ften we try to explain in simple term;; at p.'..:;ent''s meetings what we are 
rying to do. The vast majority of parent"^ l^rvc. been most cooperative and 
inder standing. I h?.vo been very plaa^sid in thir, respect, and so, I believe, 
las our administration. The best ambassador c to the home front have been 
he students themselves. From the "A'"' student on down to the "just passing" 



.# .'*»! 



- 21 



' student, there is much enthusiasm. This is due in part to the freshness with 
which the materials are written, and in part to the fact that through the way 
in which the students discover nauch of the material, they feel it is theirs 
rather than something that is being forced on them. Far less frequently, 

I practically not at all, do we get the question "Why do we have to know this?" 
Students love to be creative. The better students particularly enjoy the more 
challenging concepts. 

How do the students do on the standard tests ? We do not have much 
in the way of statistics, but we believe that on the Cooperative tests of the 
Educational Testing Service our UICSM trained students do as well as the 
traditionally taught. In the next year or so, as the tests are revised to in- 
corporate more of the newer topics, we feel our students will do much better. 
The present tests do not test the students over all the material with which 
they are familiar. Nor do they do much in testing for understanding . As for 
College Board tests, we had very fine scores this last year--higher than ever 
before. About 95% of our students are college -bound, but their ability is not 
exceptional. We get some very excellent scholars, but we have some students 
who have not shown themselves to be strong students but who go to college 
perhaps because their parents want thena to. You see, two-thirds of our high 
school population (boys and girls) consists of boarders whose parents have 
the means to send them to such a school. Our scores on the advanced mathe- 
iTaatics achievement test of the College Board were higher this year than ever 
before. For the first time we had one student receive the top score of 800, 
1 while 5 out of 9 received scores of 7Z0 or better. This class had sonne excep- 
lltional students in it, so I do not pretend that the UICSM program should take 
, all the credit, but it certainly did them no harm. 






One of the underlying principles of the-, UICSM prograna is that students 
are capable of better thinking than teachers have given them credit for; this 
is in line with the philosophy of our school: that we do not put limitations on 
'the mental capacities of the students. 

Another point we like about the program is the precision of language 
I and the use of elementary logic in the texts. The teacher is expected to use 
ithis same care in language, and, although the students aren't expected to be 
mo carefxxl themselves, they do grow to use care and to appreciate doing so. 
rl have had students tell me that they are particularly pleased with this train- 
ling, and that, in becoming aware of it, they are more precise and less 
ambiguous in all that they say and write. 

P* We had such confidence in the program that after three years we elimi- 

; nated our first year algebra course; all students, instead, begin their high 
■school mathematics with UICSM First Course. We have one exception to this: 
t^amely, one class of the top eighth grade students at our school which begins 
'irst Course in the eighth grade. 

We have found this progrann to be so successful that this year our 
classes from first grade through sixth grade are all using materials fronn 
David Page's University of Illinois Arithmetic Project. This progrann is 
ntirely separate frona the UICSM prograna, but is an outgrowth of it. 



f 



It so happens that there is a comparatively large turnover of our stu- 
tlents each year, so it is difficult to say whether students continue with mathe- 
Vmatics longer than those with conventional backgrounds. We like to think that 



iVl 



- 22 - 

' wiey do. But this transfer of students raises another problem: how do our 
students fare who transfer to other school systems after a year or two? Fronn 
what we hear, they have no difficulty. After all, after the first year they have 
acquired most of the skills conventional algebra students have learned, and 
I following the second year they have learned the Euclidean geometry. Even 
! after three years they have learned the skills taught in intermediate algebra, 
except that I find that my classes cannot reach the study of complex numbers 
I until the senior year. 

The real problem is the student who transfers to our school from 
: Other school systems. We have kept conventional classes in the 10th, 11th, 
and 12th grades for these students. The first few years we integrated a few 
of these transfer students who wanted to study, or whose parents wanted them 
to study, the UICSM math. This was not easy to do, and, as the UICSM took 
holder steps in revising their program, we stopped doing this altogether. 

This year, though, we are trying something new. We have taken the 
l>est of those 10th grade students who had a year of conventional algebra at 
other schools and formed a special class in mathematics for them. To these 
students, from September until now, I have been teaching selected topics 
from the first four units, which constitute a year and a third of the UICSM 
program at our school. These are topics they did not have or had very little 
of in algebra. Now for the rest of the year this class will study the UICSM 
Second Course. Next year they can be integrated with our other Third Course 
students. 

You should have seen the skeptical looks on the faces of these students 
jithe first week as they saw they were studying ninth grade material; since 
|ithen they have become so excited about their n;-iath that there are times when 
they can hardly stay in their seats. One girl after two weeks said: "I used 
to hate math. The teacher would do some problems and then we would do 
more like thenn for homework. It was boring. But I just love this class. I 
I really have to think. " Others have decided now to major in mathematics. 
The parents of several of these students have expressed appreciation for the 
program because of the enthusiasm of their children for it. 

1 What about those students who have had four years of the UICSM math 

and have gone on to college? We have two such groups. These students 
report that they are being exempted from certain math courses and are in 
gome instances bein;j; allowed to study honors courses. We teach our students 
no calculus. I would not call our course an accelerated course: it is more 
enrichment, giving the students solid background and real understanding. I 
asked one of my former students how well she thought she was prepared for 
her college math--having studied the UICSM course--compared to her class- 
mates at college. She replied, "Better than any of them. " The class was 
using Taylor's analytic geometry and calculus text. She said that there were 
so many questions about proving this and deriving that which the other stu- 
dents found quite difficult, but which caused her no trouble at all since she 
^ad been doing just that for four years in high school. 



The teacher has an extremely important part in the success of a pro- 
ram. The teacher's role is a bigger one in the UICSM classroom than in a 
onventional classroom--or in an SMSG classroom, as Dr. Price mentioned 
iarlier--because of the nature of the way in which concepts are carefully 



23 



i developed. It takes more work and harder study for the teacher to prepare 
for classes. I feel it is vitally important that before a teacher teach these 
materials he have had a sumnner's training in them. 



I 



I have had the good fortune to help train some of these teachers. For 
the last three summers I have been an instructor at the National Science 
Fotmdation Institute at the University of Arizona in Tucson. I have taught 
F*irst and Second Course of UICSM to teachers there and Second Course to 
teachers at the University of Illinois one sumnner. I have been interested in 
their reactions. These teachers are interested in what the program is all 
lalsout, but nnany are at first skeptical and quite disturbed as they see the 
weaknesses in the way they have been teaching, some of them for many years. 
At first they seem to resent the fact that their explanations (and those of their 
textbooks) have been insufficient and not sound mathematics. Then they have 
Ibecome really appreciative of what they are learning. Toward the middle of 
the summer as we study Units 3 and 4 they say that they would like to go back 
to Unit 1 to study it with real appreciation for what it is doing and the excel- 
lent ground work it is laying. A teacher's just reading the xinits is not enough. 
He has to see it in action and even try teaching it for one or two years before 
he can really appreciate the beautiful continuity and the wonderfully clear 
jdevelopment of concepts. 

Last summer my teacher -students who were studying Unit 5 on 
relations and fxinctions and who had studied First Course the year before 
found Unit 5 so exciting they neglected their other courses to read it. They 
nicknamed the unit the "Beberman novel" because they just couldn't put it 
down. These teachers have told me that, whether their school systems would 
Let them join the program or not, their study of these materials certainly 
affected their teaching in their conventional classes. 

Unit 6 on geonnetry teaches geometry and proof with new meaning 
.and significance. I have found it extremely enlightening. 

If you listened to last Tuesday morning's television showing of the 
series "Continental Classroom: A Course in Modern Algebra" you heard 
he speaker mention some interesting facts about how the UICSM program 
eaches nnathematical induction to its 11th grade students. In most algebra 
!)ooks--high school and college --the topic of mathennatical induction is a 
short chapter in which one gets the idea that to prove a generalization he 
Tiust assume the thing he is trying to prove. How confusing! Our Unit 7 
:arefully develops the idea of mathenniatical induction and allows the students 
o really \inderstand what mathematical induction is all about. We teach 
nathematical induction because we use it directly, and indirectly, to prove 
mr theorems about exponents and also theorems for arithmetic and geometric 
progressions. Ability to manipulate exponents is necessary, but incidental' 
O the careful development of the nnathematical concept of exponents. The 
jvay in which trigonometry is developed is also extremely clear, enlightening, 
ind sound. As you can see from my comments, I could never go back to 
leaching mathematics from traditional textbooks. These UICSM materials do 
i6 much for the teacher as for the student. 



Lest I leave you with the innpression that all is "rosy" and that this 
1$ a panacea for all the ills of teaching mathematics, let me say that this is 
t the case. Some students still don't care particularly for math, some 



- 24 - 



' students still have a great deal of trouble with it, some students never learn 
to manipulate fractions, Sonne students transfer from our UICSM program to 
our traditional, thinking that the grass is greener in the other yard; but there 
are not many of these instances, and the good reports far outweigh the bad. 

I Also, some teachers do not see the use of this program, and some outstand- 
ing naathematicians do not agree with the philosophy of the program. But it 
couldn't be much different from the traditional if everyone were in complete 
agreement about it. 

So, in summary, I feel that the UICSM program is a worthy one if for 
no other reason than that the teachers gain renewed interest in teaching nnath. 
But, the students do learn a lot nnore math, they enjoy it, they understand it, 
they do as well on tests, they become creative thinkers rather than automatic 
manipulators, and they enjoy abstractions. The more abstract the nnathematics 
is, the more practical the students tell you it is. 

vl, vl^ vl^ 

'I* '1^ '1^ 

NEWS AND NOTICES 

Sister Mary of the Angels, St. Rosalia High School, Pittsburgh, 
represented UICSM at the Philadelphia Regional Orientation Conference in 
October and appeared with Mr. Beberman as a panelist at the annual Edison 
llFo\indation Institute in Pittsburgh on November 17 and 18. 

Mr. Howard Marston of the Principia Upper School in St. Louis was 
tithe teacher-panelist representing UICSM at the Topeka, Kansas, Regional 
•Orientation Conference on December 1 and 2.-, (This Conference was omitted 
jifrom the list published in Newsletter No. 2, for which we apologize. ) 

Mrs. Teruko S. Yamaura of the Kapaa High and Elementary School, 
;Kapaa, Kauai, Hawaii, spoke on the UICSM program before some sixty 
.Rotarians who visited her school during American Education Week. She 
reports that "It was really a good experience for me, and they showed great 
interest. " 

Miss M. Eleanor McCoy, associate teacher coordinator for the project, 
returned recently from an extended visitation trip to cooperating schools in 
lawaii and some continental Western states. There are 1900 students enrolled 
[in UICSM classes in the 24 cooperating schools of our newest state. 

Mr. Beberman' s travel plans for February include the following 
ppearances and presentations: 

Tebruary 3 Lyons Township Teacher's Institute, LaGrange Park, Illinois 

8 Address the Principals of District One at Chicago, Illinois 

22 Address the Southern Regional Science Seminar for University 
Information Officers at the University of Florida, Gainesville 

24, 25 Participant in Conference on Future Responsibilities for School 
Mathematics in Chicago, Illinois 



.ion.-i: 






25 - 







NOTICE 








This issue of the UICSM Newsletter 


is the 


last to be 


mailed to 


schools 


for which we have no record of the i 


ndividuals who receive it. 


Future 


issues 


will be sent to 


named individuals 


only, so 


those who 


wish to 


receive 


further issues 


but are not 


named on the a 


ddress label nnust 


send us 


their name and 


address. A 


postal 


card will 


suffice. 





NOTICE 

UICSM will conduct a six-week sumnaer training conference in Urbana 
from June 26 through August 4, 1961. There will be classes for those pre- 
paring to teach any of the four courses in the UICSM program. We have 
received a grant from the National Science Foundation which will enable us to 
pay a stipend and allowances to participating teachers. 

Administrators of cooperating schools have been sent information on 
applications, course credit, housing, stipends, and allowances. Teachers 
in cooperating schools who might be interested in this program should consult 
with their principal or superintendent about applying. Other teachers using 
UICSM materials may wish to write to us for information and application forms. 

" V.'e know of three other institutions with summer offerings which will 

, prepare teachers of UICSM first or second course. For information, write 
I to the NSF Summer Institute Director, c/o the Department of Mathematics, 
at one of the following: 

Sacramento State College 
Sacramento 19, California 

Wayne State University 
Detroit 2, Michigan 

V/estern Washington College of Education 
Bellinghann, Washington 



-l."' 





ews dVdv 



An occasional publication of the 
UNIVERSITY OF lU^INOIS COMlvHTTEE ON SCHOOL MATHEMATICS 

1208 West Springfield 
Urbana, Illinois 



Operations as Functions 
UICSM-NETRC Tests I-L 
The New Units 7 and 8 
News and Notices 



Z 

20 
30 
32 



UICSM Newsletter No. 4 



March 27, 1961 



• ■■ . i ' . 



;,.•< V 



OPERATIONS AS FUNCTIONS 



Many of the concepts in Unit 5, Relations sind Functions , can be readily- 
anticipated in Units 1-4. [And, this can be done without using the words 
'relation' or 'function'.] Here are some ideas about how to do this. [We are 
assuming that the reader is familiar with Unit 5.] 

Let's start with Unit 1, Near page 1-7, we can make our first contri- 
bution to a better understanding of; 

(1) a relation is a set of ordered pairs; 

(2) a function is a set of ordered pairs [relation] no two of 
which have the same first component. 

'I- 



Consider a road which begins at A. 




Teacher: What is the starting point of a trip whose measure is *2 [or, 2 to 

the right, or 2] ? <; 

Student: A, 

Teacher: What is the ending point? 

Student: C. 

Teacher: Is the trip from A to C the only trip whose measure is ^Z"? 

Student: No. The trip could begin at B and end at D. 

Teacher: Any other trips of this kind? 




Teacher: 



Student : 



Teacher; 
Student: 
Teacher : 



We're getting a lot of these now. To keep from getting them 
mixed up, let's use parentheses. 

A trip beginning midway between A and B and ending midway 
between C a:id D has measure *Z. 

















__^ 


R 


eT 


-5 

c 


V^ 


F 


& 


H 


i 


(A 


c) 




P) [^ 


.«) 


(•^ 


.t) 





How many trips are there with measure *Z? 

Lots of them , 

So all these trips 3.nd inany, many more have measure *2. 



Note the "balloon' notation introduced here. 
This anticipates page l-68c. Also, see the 
discus Gipii on TC[l-l]a and TC[l-l]b. 




Teacher: Is there a trip whose naeasure is *Z and which ends at F ? 

Student: Yes, the trip from D to F. 

Teacher: Does the trip from E to C belong to this set? [The child will 

understand the word 'set' here without any explajiation. ] 
Student: No. 

Teacher: Does the trip beginning at A and ending at D belong to this set? 

- 3- 



Student: 
Teacher: 



No. 



[Pointing to '(A. C)'.] Is there any other trip besides this one that 



starts at A and has naeasure ^ ? 



It is possible- that some student may contend that the 
trip from A to C made today is different from the 
trip from A to C rriade yesterday. If this happens 
make clear that the Aword 'trip' is being used as the 
person .vho says, "I've made the trip from Urbana 
to Chicago nia-ny times",, is using it 

This development lays the groundwork for defin- 
in'' a real nuir.ber ".i; a certain set of ordered pairs 
of numbers of arithmetic. If we consider the road to 
be a nunber ray of arithmetic, we have defined *Z to 
be the set of ordered pairs ox numbers of arithnaetic 
such that the second ccnnponent of each ordered pair 
is 2 greater taan its first component. 

Now, what can we do -.vith section 1> 02, Addition of real numbers ? 



Teacher: Mary, thinh of a ,:rip on •':his road. What reed number measures 

the trip you are thinking about? 
Student: *4. 

Teacher: Beginning at the cnd-pcint of that trip, take a trip whose measure 

is *3, What is the nneasure of the single trip that would take you 

from the beginriing point of the first trip to the ending point of the 

second trip? 



As each numrer is given.- 


write a numeral for it in 


the proper place.- jjutting in the coinm.as and paren- 
theses as yon write. The stages by which you would 
arrive at '{{%, '3), 'lY are: 


""4 


*7) 


((n^-5)/7) 


_. <. V- ' >- ' — 


((-8,-^\-(o) 


((+7 -10), -3) 



-4 



" T 



¥ 



Student: *7 
Teacher: Correct. 

Continue the questioning in order to obtain additional ordered pairs whose 
first components are themselves ordered pairs. 

((^4, ^3), ^7), ({-5, ^2), -3), ((-8, -2), -10), ((^7, -10), -3) 

((^3, *4), ^7) 



[Naturally, we are not suggesting that this ordered pair notation should re- 
place the conventional '^4 + ^^3 = ■^' type of listing of addition "facts".] 

Again we have a set of ordered pairs, no two of which have the same 
first component. A function of this type in which the first component of each 
ordered pair is itself an ordered pair is often called a binary operation . 
Contrast this with: 

{(% ^7), (-3, 0), (-5, -2). {^8, ni), (-2, n), } 

This operation, adding *3, is called a singulary operation on the set of real 
numbers. 

Some mathematicians use the phrase 'operation on a set S' only if the set 
is closed with respect to the naapping. Thus, a singulary operation on S would 
be a mapping which takes you from a member of S to one and only one member 
of S. In the case of a binary operation on a set S, the mapping would take you 
from any member of S X S to one and only one member of S. In UICSM 
courses, we use the word 'operation' as synonymous with 'function'. So, for 
example, we talk about the absolute value operation which takes you from the 
real numbers to the numbers of arithnnetic. This is not a usage of 'operation' 
which conforms to the definition mentioned above. 



Part of the following is a modified transcription of the questions and 
answers given in one of the classes at University High School. Some of you 
may recall it from one of our training films. 

The day before this discussion, the class worked exercises designed to 
create an awareness of the existence of the inverse of an operation. For 
example, among them were exercises such as "If you want to undo the result 

of adding 15, subtract from the sum, " Now the class is ready to find 

out that an operation is a set of ordered pairs and then to find how to form 
the inverse of an operation. The terminology 'converse of an operation' can 
also be introduced her e, but this is optional. 

Teacher: Suppose a third-grader said to you, "What do you mean by 
'adding 9'?" What would you tell him? 

- 5 - 



,; I; 



ii. .ic 



,('i: 



• ;.". 



*.c .••' .;? 



.) ijii 5'.: 
■It i'\ 



'■^■\ V 



i i.jiiV'i' . iJi'sJ Ji 



^.,M- 



1ii:i i'i' 



: \ - I • 



-i- .;.«;< J ;••''' ^^AiSJ'.. ...i. 



U."- 



•A* 



,^ ;,/ij > f .'._ :~r -^.^^ ne-j > 






.at..'.': 



■■■•.U-.'i^'- ".V . 



iiCi.iiti,;-; ■ 
:.fH v.fjj'i- • 






1 1 



/cH '' wr.'. 



.iJ^•r..<^^;.■ 



••iJ ": 7i.;T' .'o I .'• . > 



Student: 

Teacher : 

Student: 

Teacher : 

Student: 
(Mary) 

Teacher: 
Student: 
Teacher: 
Student: 
Teacher : 



Well add a number 

That wouldn't be very helpful. 

Well, adding nine ones to whatever you're adding. 

I don't know whether that would help him or not. 

Well, if he had one apple and you gave him nine more apples, he 

would have ten apples. 

I see. What would you say, Jack? 

Uh, have him a quantity so much more. 

Joan, what do you think? 

Well, he ought to know it. 

Let's see. Mary, you said to give him an example. He has one 

apple, give him nine apples. He now has ten apples. 




What else would you do? Give hinn another example? Harry. 
Student: You could say you have 12 apples. Then someone gives you nine 
more apples. Let him count them up to see how many he has. 




Teacher: You could keep on giving him example after example. I think 

pretty soon he'd form some idea of what adding 9 is. Let's put 
some more examples down like these. 



Is'- 



-rl 



! J I.!: 



1 4<5 ^ 


lO 


11 + ^ - 


Zl 


^\ -^^ -- 


4-0 


7+9 -- 


1 (^ 


3o-)-^ = 


39 


1049 - 


■ 1^ 


2 00 + ^ 


r Z09 


Z+^ 


^ 1 1 


349 


z \Z . 



Teacher: Imagine, for a moment, that we have all possible examples of 
adding 9. How many would there be? 

Student: Lots of them. But he won't know how to add 8. 

Teacher: Then we'll make hinn a specialist in adding 9. Imagine that we 
have all possible exannples for adding 9. How do we solve a 
problem in adding 9 ? Innagine we had a book full of these exam- 
ples of adding 9. Then we gave hinn a problem: 

6 +9'= ? 
What would he do? 

Student: Look in tiie six's colunin. 

Teacher: Look for the exanaple where a '6' appears in the first column. 
Suppose he finds it. What would he find in the last column? 

Student: 15. 

Teacher: So, he knows the answer to this problem must be 15. Let's pre- 
tend that we're actually going to have a book like this--all full of 
examples of adding 9. Now, I want to save sonne space in the 
book. Is there any way in which I cam shorten these sentences so 
I can save space? 



7- 






J. . "^ * U I ' ^ '•' *. .J-' ;'/ .' V I . . ^ . J 



<;-..n 



The students made various suggestions about how to shorten these sen- 
tences. Their suggestions amounted to something like the following: 

"We don't need all those plus signs. Let's take them out. " [The plus 
signs were all erased.] Someone else sayS; "Why not remove all the equal 
signs?" [Equal signs were erased.] Finally, someone says, "Why not take 
out all the '9's, since they are repeated in each sentence?" The teacher 
then pointed out that they needed sonnething to separate the umerals for each 
pair; so, commas were decided upon. Finally, the teacher said, "Let's put 
the whole works in parentheses. " The following shows different stages of 
the development. 



1 


9 


= 


10 


12 


9 


= 


21 


31 


9 


= 


40 


7 


9 


= 


16 


30 


9 


= 


39 


10 


9 


= 


19 


2 


9 


= 


11 


3 


9 


= 


12 


6 


9 


= 


15 



1 


9 


10 


12 


9 


21 


31 


9 


40 


7 


9 


16 


30 


9 


39 


10 


9 


19 


2 


9 


11 


3 


9 


12 


6 


9 


15 



1 


10 


12 


21 


31 


40 


7 


16 


30 


39 


10 


19 


2 


11 


3 


12 


6 


15 



Recopy to save space 



1 


10 


12 


21 


31 


40 


7 


16 


30 


39 


10 


19 


2 


11 


3 


12 


6 


15 



( 1 . 


10) 


(12 ', 


21) 


(31 , 


40) 


( 7 , 


16) 


(30 , 


39) 


(10 , 


19) 


( 2 , 


11) 


( 3 , 


12) 


( 6 ., 


16) 



( 1, 


10) 


(12, 


21) 


(31. 


40) 


( 7, 


16) 


(30, 


39) 


(10, 


19) 


( 2, 


11) 


( 3, 


12) 


( 6. 


15) 


- 


- - 



Teacher: Imagine then, these are just a few sample pairs from this book. 
What would be a good name for the book? Suppose we actually 
got real silly and printed a book like this. What would be a good 
name for the book? 'David Copperfield' ? What would be a good 
name, John? 



- 8- 






jn*J 






■»'. 



I ^ii 






'. i '. 






;.-Ji 



...-f 



:'^ » .^na 



.vt..c'- 



Student: Adding Nine. 

[Other suggestions,] 
Teacher: Let's pick that short title: Adding Nine. 











/ (12,21) \ 
/ (3 1,4-0) \ 

(3o,3^) 

C/o, 19) 

( Z) M) 
(3, 12.) 

I C^, 15) / 





Teacher: This book has lots more pairs. Let's get a few more samples to 
see that we've got the idea, A pair that begins with 15, ends with 
what? 

Student: 24 

Teacher: A pair that begins with 40 ends with what? 

Student: 49 

Teacher: I'm going to write a pair and I want you to tell me if it really could 
conne from this book. Ready? (17, 26). Yes? No? Class? 

Class: Yes 

Teacher: (39,30) Yes? No? Class? 

Class: No. 

Teacher: What about this pair, (39, 30) ? What book could that come from? 

Student: The Book for Subtracting Nineo 

Teacher: I think I like that.. 

Student: The Book for Adding Negative Nine. 



O.t ■ii-^i'if''!!ti;r ••-•lA,*''.' 



,ifr!T o/ .^fc . : I .'&[••'!!■ ::: 



• '-' •* ••; J 3 < 



■tlSr.-X 



V Ji;i3 



5>s!D 



; i.-r 



rjff ;.:' : 



Teacher: Good; but remember we were talking about numbers of arithmetic. 
[Third-grade. ] 



a<Mim^ 



3 



(12, ^0 
(7, i<^) 

(^, I!) 

(4; 15) 



( 1^5,24) 
(40 ,4^) 



-^ i'^Jri}i,c,^c;tc/YU> 9 




Teacher: Let's get some more pairs that would belong to the book, 

Subtractinp; Nine, I want someone very quickly- -without doing 
much thinl:ing--to give me a whole bunch of pairs that come 
from this book. Are you re'ady? 

Student: (69, 60) 

Teacher: Too much thinking. 

Student: (10, 1) 

Teacher: Faster. 

Student: (21.. IZ), (15, 7), (39, 30). 

Teacher: You already have (39, 30). [(39, 30) was the first entry in this 
book. J 

Student: (19, 10), (209, 200), (11, 2), (12, 3). (15, 6), (24, 15), (49, 40), 

(26, 17), 

Teacher: Are there any more pairs that belong to Subtracting Nine ? 

Student: Lots. 



- 10 



^vrjwl it'^.^rVi: 



;i./><. f.: V-. ' 'V.' 



•I->JJ 



.(-..l ,.w^. 



Teacher; 



If you have this first book, do you need the second book? 




Student: 
Teacher : 



Student: 

Teacher ; 

Student: 
Student: 
Teacher; 



O 



Cl,£<£iy7oa 



9 




'^u-fi^Via/cXi^u^ 5' 




(^9,30) 

(2.1 , \'L) 
(Ifo, l) 

(n, 10) 

(209,2-C'O) 



(24 J?,) 
(49, 4o) 
(2(i.,n) 



No. 

Right. It turns out that this second book is not really necessary. 

Any problenn you want to do using the second book could have 

been done by using the first book. 

For that matter, you could say that the Adding Nine book was not 

necessary. You could use the Subtracting Nine Book. 

Right. But, suppose that we all know how to add 9, does that 

automatically tell us how to subtract 9? 

Yes. 

Would there be books like these for multiplying and dividing? 

Well, I think so. Let's nnake up another book. Here are some of 

the pairs that belong to it. 



^:UUl. 




('.'^) (15, Z4) 




11 



/iff c;r ■•■V . - .. 



Ion ' 



-fu/I -?'•:■':. A 



?i'jii 



Teacher: Give me some more pairs that belong to this book. 

Student: (400, 1600), (80, 320). (9: 36) 

Teacher: What is a good name for this book? 

Student: Multiplying by 4, 

Teacher: Remember that we could do without the Subtracting Nine book if 

we had the Adding Nine book? Is there any book we could do 

without if we had che Multiplying by 4 book? 
Student: Dividing by 4„ 

Teacher: Tell me some o;c the p?.irs that would belong to that book. 
Student: (36, 9).- (320, 80), ;i600, 400), (12, 3), (4, 1), (0, 0), (28, 7), 

(20, 5). 



\CU<S^'^ 




B^,'^:^L.'^ .6^ 4- MuiUf^^^ ^ 




(3>zo,80) 

{{(oOO-, 4-00) 
()Z,3)(28,7) 




Teacher: Does anyone set hovs- he jets the pairs that belong to this book? 

Student: Well, he just flips them around. 

Teacher: Right, you reverse the pairs. That's the way it works. Reverse 

the pairs. Is there any pair thaj belongs to both of these last two 

books ? 
Student: (0, 0) 

Teacher: Is there any pair that belongs to Adding 9 and Multiplying by 4? 
Student: (3, 12) 



i; 



Teacher: Let's look at another book. 




Student: 
Teacher: 



What are some other pairs that belong to this book?. . . 

What is a name for this book? 

Multiplying by 0, 

Let's reverse tliese pairs. What pairs will we have then? 




i^/uc<£^ rr^^MfJl^^ O Bw^JU^ iU 4-1 



(0,7) (0,8) 

(0,3) (O, IS) 

(0,0) 

(o,7Z.) (0,9O 




yy YiuJd^u^^^ ^ o 



("^><^) (8,0) 
/ (^,0) 



- 13- 



Teacher: "Which book would you use to do this problem? 



74 + 9 = 
98 - 9 = 
15 X = 



What kind of problem could you work using this book? 
[Pointing to the unlabeled balloon.] 
Student: There isn't any. You'd never know which pair to pick because 

they all start with 0. 
Teacher: Right, Then can you divide by ? 
Student: No. 

Teacher: Can anyone divide by ? 
Student: No. 

Teacher: However, there is a name for this last book. It's called 'The 

Converse of Multiplying by 0'. Converse. C-O- N- V-E-R-S-E. 

Can you give me another name for this book? 




Student: The Converse of Adding 9. 

Teacher: Correct. Let's put that down. 



14 



Student: I think it ought to be called "The Reverse of Adding 9". 

Teacher: Maybe that would be a good name, but that's not what mathema- 
ticians call it. "We reverse the pairs and we get the converse. 
"What is another name for the book Adding 9? 

Student: The Converse of Subtracting 9. 

Teacher: [Writes this in,] What is another nanne for the book Dividing by 4? 
Multiplying by 4 ? 



a — . 



] [ 



(^^i-dc<iM^^^ ^ 







hi^/-~£l^.^ i>-M 4- 



I "7 ksL '^jpA^^MyiSi. i/ 



n\M:f.^^ii^i- 




ylhs- Cif/iy/^yUiSi^ of 






muiltojc%:.^lL 






o 





Students may want to give another name to Multiplying 
by 0. They may give 'The Converse of the Converse 
of Multiplying by 0. ' This is acceptable. But, of 
course, 'The Converse of Dividing by 0' is meaning- 
less and not acceptable, since 'Dividing by 0' has no 
referent. 

Teacher: Notice that in this book, Subtracting 9, if we start with 15, we go 

to 9 and nowh ere else . In the book, Dividing by 5, if we start 

with 35, we go to 7 and nowhere else . Is this sort of thing true 

for the book we call 'The Converse of Multiplying by 0' ? 

Student: No. 

- 15- 



li .i" ':■. 



Teacher: Is it true for the book, Adding 9? Multiplying by 4 ? Multiplying 

by 0? 
Student: Yes. Yes. Yes. 

Teacher: Books like this, where you go nowhere else , we call 'operations'. 

Is Dividing by 5 an operation? 
Student: Yes. 

Teacher: Which of these books are operations ? Which are not operations ? 
Student: All but one are operations. The Converse of Multiplying by is 

not an operation. 
Teacher: Each of you make up a book which is an operation. Draw a loop 

on your paper and list five pairs which belong to that operation. 

Give your operation a simple name. 

[Check this work some way at this point.] 

Now draw another loop. Reverse the pairs that are listed in your 

first loop and list these reversed pairs in the second loop. Is 

this set of pairs an operation? Give this set of ordered pairs a 

name. "s 

The word 'ordered' can be used here without special 
emphasis. If students question this, just point out 
that you are saying 'ordered' because it makes a 
difference about which number comes first. 

Teacher: We have a special name if both the book and its converse are 

operations. We use the name 'Inverse' instead of 'Converse'. 

Inverse. I-N- V-E-R-S-E. Now think about the book, Adding 9. 

Think about the book, The Converse of Adding 9. What other 

nanae have we already given this second book? 
Student: Subtracting 9. 

Teacher: What new name can we give this book? 

Student: The Inverse of Adding 9. [Write this name in the proper place.] 

Teacher: What other name can we give the book Adding 9? 

- 16- 



V 



. . I .■ ^ "f ■ <r •• 



!•'■ -.V 



■fi:i 



Student: 

Teacher : 

Student: 
Teacher : 
Student: 
Teacher : 



The Inverse of Subtracting 9. [Write this name in the proper 

place.] 

Is there any one of these books for which we will not use the word 



inverse 



Yes. The Converse of Multiplying by 0. 

Will we use the word 'inverse' for any other of these books? 
Yes. Dividing by 4 is the Inverse of Multiplying by 4. 
Any others ? 



^.Jb^bi^. 



■-<lt.,,A^ '^1 







8wJU.^A-^ i6-74n 












\ Tn i/iiX-|fLkAJ /^ X/"-\ ^ 







^^A- 




7 ^-£. C<ryu-%KyT^i-2- (^ 








Teacher: If the word 'converse' appears in a name, can we always replace 
the word 'converse' by 'inverse' and get another correct name? 



17 



\ t 



I ( 



r 



■I 'djiti'. 1 



J 



Student: No. 

Teacher: If the word 'inverse' appears in a name can we replace the word 

'inverse' by 'converse' and get another correct name? 
Student: Yes. 

Teacher: If we can correctly use the word 'inverse' we usually use it 

instead of using the word 'converse'. V/ e would seldom say: 
Subtracting 9 is the converse of adding 9. 

"We would usually say: 

Subtracting 9 is the inverse of adding 9. 

Now look at your papers. Write another name for each of the 

sets of pairs. 

If the word 'converse' appears and if the word 
'inverse' is applicable, ask them to write still 
another name. 

Here are two sets of pairs. 






(^,4-) (7,11) 
(1,5) (^,13) 

(iej4,) 



Student: 
Student: 
Teacher : 
Student: 
Teacher : 



[Ask them to give more ordered pairs for each operation.] 
What is a name for this first operation? For the second? 
Adding 6. The Inverse of Subtracting 6. 
Adding 4. The Inverse of Subtracting 4. 
Now. Jane, pick a nunnber. What one did you pick? 
7. 
Let's go to the first operation. Adding 6. ' . . 



18 



::«i-M!. 



. i. !- L-.l ■ 



.( Oi^ 



[Put '(7, )' inside the proper loop.] 

Teacher: Perform this operation on 7. What second number do we get? 

Student: 13 [Write '13' in the proper place.] 

Teacher: Now perform this second operation, adding 4, on the number 13. 
[Write '(13, )' inside the proper loop.] What shall I write? 

Student: 17 

Teacher: So, if I start with 7 [write '(7, )'] and perform the operation, 

adding 6, then take that result and perform the operation adding 4 
on it, I get 17. [Write '17' in the proper place.] 




,i>Ui^I^-V^-6, 



4- 



(2,0) Co,(.)\ 
(3,9) (4-, 10) 






n,n) (2,iz.": 




Teacher: Let's pick another number. 

Student: 2 - 

Teacher: Perform the first operation. Now perform the second operation 
on that result. What do you have ? 

Student: 12 

Now have each child go through this procedure. List the 
ordered pairs as shown above. Draw a loop around them. 

Teacher: How many such pairs are there? Imagine that we have all such 

pairs. Do we want to say that this set of pairs is an operation? 

Student: Yes. 

Teacher: Can you suggest a good name for this operation? 

Student: Adding 10. 



The examples given above have been based on the set of numbers of 
arithmetic. Examples using real numbers and the operations addition and 
multiplication on real nunnbers can also be used. 



- 19- 



S,t<}'f.-; 



■M- ::<y<~ii- 



UICSM-NETRC Math Study Tests 



In Newsletters 1 and 2 we presented the first eight tests in the UICSM- 
NETRC series. Here are the next four, designated by the letters I, J, K, 
and L. The page of the text to have been conapleted is again in brackets 
after the letter. A fifteen minute time linnit has been set for each for the 
purposes of the NETRC Math Study. --R.S. 

Test l[2-59] 

I. Some geometric figures have been drawn and the relative measures of their 
sides indicated with pronumeral expressions as below. Choose the expres- 
sion which, when written in the blank in 'P = ', would yield a correct 

perimeter formula for the figure. 



1. 





/^ 


\^ 




I 




2. 


a 




a 










s 






a 






(A) 


sss 




(B) 


3s 






(A) aaaa 


(B) 


aa 




(C) 


s 
3 




(D) 


none 


of these 




(C) 


aa + . 


aa 


(D) 


none 


of these 




4. 




(A) aa + b 
(C) aab 



(B) 2a +b 

(D) none of these 



(A) abc 

(C) a + b + c 



(B) 3b 

(D) none of these 



X + 3 



5. 



X + 3 
(A) X + 6 (B) xxxx + 6 

(C) XX + 6 (D) none of these 



6. 



(A) 3x + 10 

(C) 5x + 10 




X + 7 



3x + 3 

(B) 3xxx + 10 

(D) none of these 



7. 



3X 




|(x + 2) 



X + 1 



(A) 2x + 3 

13 
(C) -T-x + 2 (D) none of these 



(B) ^x + 3 



8. 



x + 3.2 




2x + 0. 3 
(A)5. 2x+3.5 (B) 2.4xx + 1. 86 

(C) 3. 2x + 3. 5 (D) none of these 



20- 



-. ■!•■ 



•r.q 



>r. ' 



'•) 



.,r.:i: ^.^ 



>' i 



II. Choose the equivcilent pronumeral expression from among those listed, 
if there is one. 

9. 2a + 3b 

(A) Bab (B) 6ab (C) 5(a + b) (D) none of these 

10. x(3xy)(5x) 

(A) 9xy (B) 15xy (C) 15xxxy (D) none of these 

11. 3(3 +y) + 7(y + 3) + 3 +y 

(A) ll(y + 3) (B) 9y + 15 (C) 10(yyy + 3) (D) none of these 

12. (3x)(^y) + (-^x)(-|y) 

11 5 

(A) ^xy (B) -rxy (C) -xxxyy (D) none of these 

13. y(y +3) + 3(y +3) 

(A) 3y(y + 3) (B) (y + 3)(y + 3) (C) yy + 3y + 6 (D) none of these 

14. XX + 3x + 2 

(A) x(x + 5) (B) x(x + 3x) + 2 (C) (x + l)(x + 1) (D) none of these 

15. a(2a + 3b) + b(3a + 2b) 

(A) (a + b)(2a + 3b) ^ (B) 2(aa + 3ab + bb) 

(D) (a + b)(3a + 2b) (D) none of these 

16. XX + 2xy + yy 

(A) (x +y)(x + y) (B) x[(x + 2y) + y] 

(C) x(x + 2)y(2x + y) (D) none of these 



III. Choose the correct completion for a true generalization. 

17. For each x, the sum of (3x + 2) and the product of 4 by {5x + 3) is 

(A) 17x +11 (B) 23x + 14 (C) 60xx + 6 (D) none of these 

18. For each x, for each y., the product of the sum of (x + y) and y by the 
sum of (2x +y) and (y + x) is 

(A) (2x + 2y).(x +y) (B) 3xx + 4yy 

(C) (x + 2y)(3x + 2y) (D) none of these 



21 



,ft-i ;-' 



;)r?fi-! 'C 



jje^'iiy: 






it' 



; f 

.C) 



•.U 



IV. Choose the correct reason for the numbered step. 



(2x + 3y)(x +y) 
= (2x + 3y)x + (2x + 3y)y 
= 2xx + 3yx + (2xy + 3yy) 
= 2xx + 3yx + 2xy + 3yy 



19. 



20. 



(A) apna 
(C) Mpma 
(A) cpna 
(C) cpa 



(B) dp ma 

(D) none of these 

(B) apm 

(D) none of these 



Key for Test l[2-59]: 



1. B 


2. D 


3. B 


4. C 


5. D 


6. C 


7. 


C 


8. A 


9. D 


10. C 


11. A 


12. A 


13. B 


14. 


D 


15. B 


16. A 


17. B 


18. C 


19. C 


20. D 






Test J[2- 


■80] 















I. Which of the given expressions, if ajiy, will yield a true generalization 
when written in the blank? 



1. V^-{3 - x) 
(A) -3 - X 



(B) X- -3 



(C) X - 3 



(D) none of these 



2. V V 13x - 5y - 7x + 3y = 

X y ' ■' 

(A) 8xy - 4xy (B) 6x - 2y 



(C) 6xx - 2yy (D) none of these 



3. Vj^V {3x - 4y) - (y - 2x) = 



(A) 5(x - y) 



(B) X - 5y 



(C) 5x + 3y 



(D) none of these 



4. V V x - y(x - y) 

X y ' ' 



(A) X - (x + y)y (B) x - xy - yy (C) x - xy + yy (D) none of these 



^' ^x\^z X - (y - z + x) = 

(A) 2x - y - z (B) - (y + z) 



(C) z - y 



(D) none of these 



6. V V 3(2x - 3y) - 5(x - 2y) 



(A) X - y 



(B) X + y 



(C) X - 19y 



(D) none of these 



7. V^V (x + y)(x - y) = 

(A) XX - xy + yy 
(C) XX + xy - yy 



(B) x(x - y) + y(x - y) 
(D) none of these 



-22 






:fi*'ri^:ji.^-.:^^^; 



ike it 



■ '■i: 



■' . . I 



'^' ■ ' irfl^.i (0:i 



•;-5) 



.0? lo ii-'.^f. f.^l.; 



5;<-i -Vx 



{*!J 



:•) 



Cm 



•^•:^';1:^ t,.. -iAv-':'. 



^.;-:' .'i^s iv. ^.'*C- *■;■• U'.>L'i 



/«! ■♦■ \'>: 









• .,■•.•. 

■1 ■" X., '. *-.' 



-'.S:^- 






(A) XX +yz (B) (y - x){z - x) (C) xx - yz 



(D) none of these 



9. V V x(x + -y) + -y(x + -y) 
X y 

(A) (x - y)(x - y) 

(C) XX - xy - yy 



(B) XX - 2xy - yy 

(D) none of these 



10. V^ (x - l)(x - 2) = 



(A) XX + 2 



(B) XX - 3 



(C) XX + 3x - 2 (D) none of these 



II. What is the correct reason for each step in this proof of: 



V^Vy -(xy) = -xy 



xy + — xy >, 



= {x + -x)y <^ 



Oy 



< 



= yO 



= 

Hence, xy + — xy =0 
So, -(xy) = -xy 



11. 


(A 


1 ps 




(B 


) po 




(C 


1 idpma 




(D 


1 none of these 


12. 


(A 


> V^V -xy = (-x)y 




(B 


) po 




(C 


I ps 




(D 


1 none of these 


13. 


(A 


1 paO 




(B 


cpm 




(C 


pmO 




(D 


none of these 


14. 


(A 


> V Ox = 
x 




(B 


1 Vy y + = y 

I V = zO 
z 




(C 




(D 


none of these 


15. 


(A 


) po 




(B 


V^Vy -xy = x(-y) 




(C 


0-sum theorem 




(D; 


none of these 



- 23 



v* y - 



III. What is the correct reason for each step in this proof of: 



V^Vy V^ x(y - z) = xy 



xz 



x(y - z) -X 



= x(y + -z) 



= xy + x(-z)">. 



= xy + -xz K 



= xy + - (xz ) -< 



= xy - xz 



J 



16, 



17. 



19. 



20. 



(A) po 

(B) ps 

(C) V^Vy x - y = -(y - x) 

none of these 



(A) dpma 

(B) idpma 

(C) V V V x{y - z) = xy - xz 

X y z •' ' 

(D) none of these 



(A) V^V„ x(-y) = -xy 

X y 

(B) V^V x(-y) = -(xy) 

X y 

(C) V^Vy -xy = -(xy) 

(D) none of these 



(A) V^V^ -xy = x(-y) 

X y 

(B) V V -xy = -(xy) 

X y 

(C) V^V^ -(xy) =x(-y) 

X y 



(D) none of these 



(A) V V X - y = X + -y 
X y 



(B) po 

(c) v;; 

(D) none of these 



(C) V^V -xy = -(xy) 



Key for Test J[2-80] 



1. 


C 


2. 


B 


3. 


A 


4. 


C 


5. 


C 


6. 


B 


7. 


B 


8. 


B 


9. 


A 


10. 


D 


11. 


D 


12. 


B 


13. 


B 


14. 


C 


15. 


C 


16. 


B 


17. 


B 


18. 


A 


19. 


B 


20. 


A 







-24- 



■.<>< - <••< 



( -., 



^ • '\- 



.' ' . 



Test K[2-85] 

1. Which of these operations does not have an inverse? 
(A) sameing (B) oppositing 
(C) absolute valuing (D) adding 

2. Which of these operations does not have an inverse? 
(A) nnultiplying by (B) adding 1 

(C) oppositing (D) multiplying by 1 

3. Which kind of problem always has an answer? 

(A) division of real numbers (B) division of numbers of arithmetic 

(C) subtraction of real numbers (D) subtraction of numbers of arithmetic 

4. These pairs belong to certain operations. Which operation does not have 

an inverse ? ' 

(A) (2, 4), (-2, 4) (B) (2, 5), (5, 8) 

(C) (0,-5), (5, 0) (D) (7, 7), (-3,-3) 

5. Which of these generalizations is f al s e ? 

(A) V X = X (B) V V if X = y then y = x 

X X y ' ' 

(C) V V V if X = y and y = z then x = z (D) none of these 

X y. z ' ' 

6. Which generalization states that subtracting is the inverse of adding? 
(A) Vj^V (x + y) - y = X _ (B) V^V (x - y) + y = x 

(C) V V X + -y = X - y (D) none of these 

X y 

7. Which of these expressions is not a numeral? 

'^•oT-3 <^'|4| (C,l^ ,D,^ 

8. Which of these generalizations was not a basic principle? 
(A) paO (B) pmO (C) pml (D) po 

9. A new operation, indicated by a 't^t', is to be carried out in one of the 
ways shown below. In which case will it be commutative? 

(A)x-i!^y=x + y+xy (B)x-!i-y=x+y+y 

(C)xiiry=x+yy (D)x'j!ry= xxy 



- 25- 



•vv>.(i : 



:.GO n; 



\'' 



X •■ - .f.. -u 



■: :i>t:it- ' '■ ■ -it '■:!. '■ 



.■^fffl ..' 



. J •'.; 



'•'iii 



>i:' 



■^) .^} 



ssr ?,-{'. -^i 



.•••/,■> 



10. A new operation, indicated by a '?!<', is to be carried out in one of the 
ways shown below. In which case will it be commutative? 

(A) X '!< y = (xx + yy) v (x + y) (B) x 'i^ y = xx -^ yy 

(C) x '1- y = XX - yy (D) x ']- y = (x - y) - (y - x) 

11. For each real number x there is a real number x* such that x + x* = 0. 
Then, it is not the case that, for each x, 

(A) X - X* = 2x (B) XX* is positive 

(C) XX > XX* (D) x*x* is nonnegative 

12. For each nonzero real number x there is a real number x such that xx = 1. 
Then, for each nonzero x, x is the same as 

(A) 1 (B) 1 -f X (C) -X (D) 



13. Which of the following is a true generalization? 

(A) distributive principle for addition over subtraction 

(B) distributive principle for subtraction over addition 

(C) distributive principle for subtraction over subtraction 

(D) none of thena 

14. Which of the following is not a true generalization? 

(A) distributive principle for absolute valuing over addition 

(B) distributive principle for oppositing over addition 

(C) distributive principle for oppositing over subtraction 

(D) distributive principle for sameing over subtraction 

15. For each x, for each y, ? + x = y. 

(A) X - y (B) -(x + y) (C) y - x (D) none of these 



16. For each x, for each y, x - 



y- 



(A) X - y (B) y - X (C) - (x + y) (D) none of these 



17. Which "cancellation principle" is not a true generalization 

(A^ V V if -x = -y then x = y 
X y ' 

(B) V V if |x| = |y| then x = y 

(C)VVV ifx + z=y+z then x = y 
X y z ' ' 

(D) V V V if X - z = y - z then Jf = y 
X y z ^ ' 

- 26- 



I.-- '-i ■ \ 



r. j ' ' 



■'. i\x) 



•:1 



1 (■. 



.'J 



f-k 



■H.r ■ 1-J, 



18. Which "cancellation principle" is not a true generalization? 

(A) V V if 13x = 13y then x = y 
X y ^ •' 



^^^ ^x/O^T^O if -^ = -y then x =y 
- z =y- zthenx = y 



(C) V^VyV^ifx 

(D) V„V„V„ vn if xz = yz then x 



X y z 



^0 



19. Suppose there is an operation called 'bracketing', such that, for each y, 
bracketing with y has an inverse. What else can you conclude about 
bracketing ? 

(A) it is commutative (B) it is associative 

(C) it has a cancjellation pirinciple (D) nothing 

Key for Test K[2-85] 



1. 


C 


2. 


A 


3. 


C , 


4. 


A 


5. 


D 


6. 


A 


1. 


B 


8. 


B 


9. 


A 


10. 


A 


11. 


B 


12. 


B 


13. 


D 


14. 


A 


5. 


C 


16. 


A 


17. 


B 


18. 


B 


19. 


C 











Test L[2-108] 

1. Choose the correct simplification, if it is given 
3 a 2bb 



1, 



-4b -3 
(A, if 



(B) 



(C)^ 



(D) none of these 



(A) \ 




(B) 


i 
6 


3 - + -^ 








(A) \ 




(B) 


a 
8 


4 12^u . 
2w 


3u 

V 






(A)f 




(B) 


18uuu 

WV 



(C) 



(C) 



aa 



(C) 2 



(D) none of these 



(D) none of these 



(D) none of these 



27- 



■iC;jj£r, 



...,■ (•:;; 



5. -^ + ^ 



Zrs 8rst 

t + 5 

rst '■"' 4rst '^' 8rst 



iA\ 1L±A ,„v I4t + 3 ,^, 28t + 3 .^. , ^, 

(A) — - (B) •— i -: — (C) -3 — -— (D) none of these 



6- ^hl^ ' ^Td-'h 



(A)| 




<B)i^ 




(C) 


12 


'■ ^ - t'^^ 












'->t^3 




<B) 3^^'x 




(C) 


X + 2 
2x - 3 


n 2 ^ 


5 










u. _ 

y - 7 y - 




(y- 0(y 


12 
-5) 


^^^'(7^ 


y) 


(C) 


-y- 31 
{y-7)(y-5) 



(D) none of these 



(D) none of these 



(D) none of these 



II. Choose the correct answer, if it is given. 

9. Which of the following is not a true principle? 

(A) distributive principle for multiplication over subtraction 

(B) left distributive principle for multiplication over subtraction 

(C) distributive principle for division over subtraction 

(D) Isft distributive principle for division over subtraction 

10. V/hich of these numbers is a counter- exanaple to the generalization: 

X < X 

(A) -1 (B) (C) 1 (D) none of them 

11. Suppose ';i' is given larger and larger positive values. Then what 

4 

happens to the values of ' —-z ' ? 

X + i 

(A) they increase (B) they decrease 

(C) they stay the same (D) cannot tell 

12. Suppose 'x' is given larger and larger positive values. Then what 
happens to the values of ' 7- '? 

(A) they increase (B) they decrease 

(C) they stay the sanae (D) cannot tell 

- 28- 



r ii> 



si-f jj-v jjf'-Sf; j'C ■ 



,v .' 






■i . 



.1 . 



13. Which pair of expressions are equivalent? 

(A) ^ +^; ^^4^^ (B) ^; ^ 

yx x+y yy + 1 

(C) -^ ; r-:r- (D) none of them 

3x - y y + 3x " 



14. V^^oVj^ a. = b 

(A) I (B) I (C) -^ (D) none of these 



15. V^ /n V„ /n X ^ = Y 



'x^O y 7^0 



X 



(A) y (B) - (C) -2^ (D) none of these 

y ^ 



16- ^a^^O^b^^o <^^^) ^ 



(A) - (B) ^ (C) g (D) none of these 

a aa do 



17. For each nonzero real number x there is a real number x such that 
XX = 1. Then,, it is not the case that, for each nonzero x, 

(A) X -r X - XX (B) X -r X = X -r X 

(C) X + X = (xx + l)x -, (D) XX = 1 

18. Which generadization tells you that, given a first nunnber [other than 0] 
and a second number, there is a number whose product by the first is 
the second ? 

(A) the zero-product theoremi (B) the principle of quotients 

(C) the division theorem (D) none of these 



Key for Test L[2-108] 



1. 


A 


2. 


D 


3. 


D 


4. 


A 


5. 


C 


6. 


D 


7. 


C 


8. 


B 


9. 


D 


10. 


D 


11. 


B 


12. 


A 


13. 


D 


14. 


A 


15. 


B 


16. 


C 


17. 


B 


18. 


B 















- 29- 



I "il. •■■>. ; 1 : 



THE NEW UNITS 7 AND 8 

The latest revisions of the UICSM units titled 'Mathematical Induction' 
and 'Sequences' are now in active preparation and are scheduled for publi- 
cation by the University of Illinois Press in the sunnmer of 1961. Unit 7 is 
intended to provide the textual basis for about the first half of the third year 
of study in UICSM mathematics, while Unit 8 contains material for the re- 
mainder of the year. 

The titles may be somewhat misleading in that they do not suggest the 
full range of mathematics encompassed by the units. The purpose of this 
article is to survey the structure and content of the two units, noting the 
arrangement and major topics of each. 

Briefly; Unit 7 is a revision and expansion of the material in the old 
unit on mathematical induction preceding the introduction of sigma-notation. 
Unit 8 begins with sigma-notation and includes the niaterial of the old Expo- 
nents and Logarithms unit up to the introduction of rational exponents. [The 
remaining material in the latter unit will be covered in Unit 9, whose publi- 
cation is scheduled for 1962, ] 

Unit 7, Mathematical Induction, of course presupposes Units 1 through 
6- -in particular, an understanding of the material covered in the appendix on 
logic in Unit 6 (published, along with Unit 5, by the University of Illinois 
Press last summer). 

The unit opens by posing some attention-getting problems on progressions 
and series, then presents a short section on connputational procedures in 
place-value numeration systems and algebraic algorithms for muliplication 
and division of polynonnials, designed as the review- with- extensions probably 
needed by most students at this point. 

Then the text launches into the main business of the unit: a detailed 
study of the beginnings of the "finer structure" of the real number system, 
and especiallv of the set of integers. To this end, the student is first led to 
a consideration of the inadequacy of the ten basic principles for real numbers 
developed in Unit 2 for distinguishing between positive and negative numbers. 
Four basic principles for positive numbers are then formulated and these 
are connected ■^.o the ' > ' relation by another assumption. Work on theorems 
and transformation principles for inequations follows. 

The postulates necessary for investigation of the set of positive integers 
are then motivated and three more basic principles introduced, the third 
being an induction principle that leads directly to proofs by mathematical 
induction. Proofs of theorems about positive integers are then carried out 
on the basis of recursive definitions, and applications are made to polygonal 
numbers and combinatorial problems. 

The existence of a least member in every nonempty set of positive inte- 
gers is then proved and the existence of an integer greater than any given 
real number is postulated. The postulate, labeled the 'Cofinality Principle' , 
corresponds roughly to that traditionally known as the axiom of Archimedes 
(or Eudoxus). It is needed principally for the introduction of the greatest 
integer function. 



30 



.1. i;:V: OS 



^ ivi < 



•i.n' 



.:•. :J. 



Divisibility is defined in another principle ajid the Highest Common 
Factor algorithm developed, then applied to solving some Diophantine 
problems. 

Many lists of Miscellaneous Exercises of mixed types and varying 
degrees of difficulty are interspersed throughout the unit to provide oppor- 
tunity for application of old and new concepts. These exercises also serve 
to nnaintain skills in manipulation and techniques of problem- solving. A 
variety of geometrical and worded problems are included. There is also 
an extensive list of Review Exercises near the end of the unit. 

Finally, there is appended, for easy reference, a list of basic prin- 
ciples and theorems mentioned in the unit. The list contains the ten basic 
principles of Unit 2, the additional assumption that 1^0, the 78 theorems 
of Unit 2 as given on pages TC[2-61]c through i, and the 51 additional 
theorenas proved in Unit 7. The new principles adopted in this unit are 
also given at their appropriate places. 

Unit 8, Sequences, opens with a section on sigma-notation, as men- 
tioned above, then takes up sunn.nn.ati on of sequences, both directly and by 
the constant difference nnethod. Arithmetic progressions and means are 
considered, then the Pigeonhole Principle. 

The nnultiplicative part follows, with an introduction to the pi-notation 
for products, more work on integral exponents, and developments of both 
finite and infinite geonnetric progressions. Finally, there are sections on 
factorials, further combinatorial problems, and the binomial theorem for 
integral exponents. --R. S. 



The object of a mathematical education is to acquire the powers of 
analysis, of generalization, and of reasoning. 

To teach mathematics is to teach logical precision. A mathematical 
teacher who has not taught that, has taught nothing. 

The habit of logical precision is the instinct for the subtle difficulty. 

--Alfred North Whitehead 

'The Principles of Mathennatics in 
Relation to Elementary Teaching' 
(1913) 

CORRECTION: We regret that several errors in the use of punctuation and 
quotation marks remained in our last issue despite what was (we thought) a 
conscientious job of checking and proofreading. Most of them were in the 
article titled 'v8 : Rational or Irrational?', including this major erratuna: 
the fourth line from the bottom on page 18 should have read 'integer and 2p 
is an integer. So, (8q - 2p) is an integer. ' We thank the alert and helpful 
readers who pointed this out. Corrections from readers are always grate- 
fully received, of course, though in our immediate chagrin we may feel a 
bit like awarding the prize of a one-way ticket to Zabranchburg to the one 
submitting the longest list. --Ed. 

- 31- 



NEWS AND NOTICES 

We are again pleased to note here the professional activities of UICSM 
teachers that have come to our attention lately through their weekly reports 
and letters to the project office. We hope that all UICSM teachers will keep 
us posted on their contributions to mathennatics education, whether large or 
small. 

Mrs. Mary S. Huzzard, Cheltenham High School, Wyncote, Pa., spoke 
to the junior and senior high school mathematics and science teachers of 
Jenkintown, Pa., on the UICSM courses; also to the fourth, fifth, and sixth 
grade teachers there, 

Mrs. Ruth Wong, University of Hawaii High School, Honolulu, spoke on 
UICSM in connection with a seminar of practice teachers and cooperating 
teachers in the Honolulu public schools. The seminar included a symposium 
on recent curriculum trends, Mrs. Wong also presented UICSM to a General 
Semantics group there, which, she reports, "was highly interested in and 
enthusiastic about emphasis on precision in language and nonverbal aware- 
ness. " An interesting "public relations" sidelight: she further notes that 
"There was also one individual who seemed intent on having me make sweep- 
ing claims for the program only to 'prove' I was wrong. I avoided most of 
these leading questions by refusing to give a pat 'yes' or 'no' as he seemed to 
want me to do. " 

Mr. Eugene Epperson, Talawanda High School, Oxford, Ohio, conducted 
a workshop for junior and senior high school teachers at Wilmington, Ohio, 
on February 28. He discussed the introduction of basic principles and the 
solution of equations and worded problems as per Units 1 and 3. The work- 
shop was sponsored by the Ohio Council of Teachers of Mathematics Workshop 
Consultant Service, 

Mr. William Annett of Seaford Junior-Senior High School, Seaford, Long 
Island, New York, reports he is conducting an adult education course in Units 
1-4. Fifteen parents are enrolled in the course, most of whom have children 
taking First Course at Seaford. The course was requested by parents and 
begun during the second semester in order to ". . .insure no parental inter- 
ference with discovery on the part of the children. " Mr. Annett plans to 
devote ten meetings (of two hours each) to Units 1 and Z this semester and 
continue with Units 3 and 4 during the first semester of 1961-62, then begin 
again with a new group. "These parents exhibit almost as much interest as 
the children, " he adds. 

Sister Mary Rosaria, St. Basil High School, Pittsburgh, presided at the 
mathematics section meeting of the diocesan secondary school convention on 
February 3. 

Sister Grace Marie of the Villa Maria Academy, Erie, Pa. , was in 
Pittsburgh during the week ending February 17 as a member of the evalu- 
ation committee visiting Elizabeth Seton High School. 

Mr. Richard Fleischer, Johnson Regional High School, Clark, New 
Jersey, taught his Unit 6 (lOth-grade) class as a denaonstration class at New 
Jersey State Teachers College on March 11, 



32- 



^i ill" 
.^ir^'■■■ 



•l\i b^ 



•'Mi / 



•f.-ii:. 



'•••*j) 



• U lOi'i '.•■ 



Wf>l/' 



•y 



Mr. Fred H. Green of the North Plainfield (New Jersey) High School has 
been appointed Assistant Director of the NSF- sponsored Summer Mathematics 
Institute at Baldwin- Wallace College, Berea, Ohio, this sumnner. He will 
conduct the seminar on "Curriculum Reform Proposals and Their Imple- 
mentation" in which UICSM, SMSG, and Ball State materials will be examined 
and in which he hopes to "promote plenty of discussion and discovery. " With 
respect to UICSM, Mr. Green observes that "I personally meet too many 
uninformed experts on our program. . . those who know very little about what 
we are doing, yet condenan isolated segments of the material. " 



Ivlr. Beberman traveled extensively during February and March, visiting 
UICSM and NETRC (film project) teachers, delivering speeches and lectures, 
and conducting demonstration classes in a number of communities across the 
nation. His February schedule included three engagements in Florida, an 
SMSG conference in Chicago, and visits to four locations in the state of Wash- 
ington. After several more engagements in Washington and Oregon during 
the early part of March he spent about a week in Alaska, making presentations 
to several teacher's groups there, then squeezed in several speeches and 
demonstrations in California before returning to do the sanne at a Rockford, 
Illinois, teacher's institute last week. He has just returned from another 
week-long trip and will be away on yet another next week. 

Miss Hendrix also traveled extensively in the past two months in connec- 
tion with her research on nonverbal awareness and the psychology of learning. 
She showed some of the NETRC films to several university audiences in Utah 
during February, delivering an address or conducting a discussion each time 
on the significance of the film for the psychology of learning. Similar pro- 
grams were also conducted in Oregon, California, and Indiana during 
February. She was interviewed in connection with her research interests 
over the University of Illinois television station on March 14. Her plans 
include a presentation at the annual meeting of the National Council in Chicago 
on April 7, attendance at a Washington, D. C. , conference on productive 
thinking in education April 27-29, and several engagements in New York, 
Massachusetts, and Connecticut in May- -including a research consultation 
with Margaret Mead and the staff of Seeing-Eye, Inc., in New Jersey. 

Mr, Arnold Petersen, Teacher Associate with the project this year while 
on leave from his position as Head of the Mathematics Department at the 
Pascack Valley Regional High School, Hillsdale, New Jersey, visited the 
Lakeview Public Schools of St. Clair Shores, Michigan as the project repre- 
sentative on February 23 and 24. He later visited his own school to discuss 
expansion of the UICSM program, then began work on an attempt to organize 
an NSF sumnner institute in northern New Jersey or the New York metro- 
politan area for 1962. He visited Talawanda High School, Oxford, Ohio, on 
March 3 en route to Urbana. 

Miss Eleanor McCoy visited the laboratory school at Western Illinois 
University on March 13, then spent the next two days at Pekin High School, 
where she had taught before joining the UICSM staff. She also visited five 
UICSM schools in northern Illinois during the week of March 20. 



33- 



• 'hiii \ 



•t ; ; 



'tl.* 



•fV 



v_y 1 





ewseiTer 



An occasional publication of the 
UNIVERSITY OF ILLINOIS COMMITTEE ON SCHOOL MATHEMATICS 

1208 West Springfield 
Urbana, Illinois 



Editorial 2 

Teaching First Course 3 

On Ordering Text Materials 4 

Calling All Cartographers 6 

UICSM in Grades 7 and 8 9 

News and Notices 11 



UICSM Newsletter No. 5 



May 19, 1961 



>■■)> 



jj.i 'u -"■' 



.r--f»' 



J-:- 



U 



f 

f I 



I >«r • r • ; • •' T ; -! 



•O ! 



■ I 



.f^Hir.A }.-.-/i '■.'./.••y, 



•^».^i .('i V. ••.)•/ 



.:< •• .Tjt*.jr>-^;i^ f/^ior 



A NOTE TO UICSM TEACHERS 

This is the fifth and last Newsletter to be published by UICSM during 
the 1960-61 school year. The Newsletter was conceived principally as an 
avenue of communication from the project office to you, the teachers using 
UICSM text materials. We have tried to keep it as practical and useful as 
possible, thinking this might best be done by publishing materials that would 
supplement and amplify the Teacher's Commentaries, 

Your response has been gratifying and encouraging. Our hope and 
intention is to continue to meet your needs in any way possible. You can 
help us do this by continuing and increasing the "feedback" to the UICSM 
office in the form of letters containing comments, pedagogical suggestions, 
expository articles, test items, and accounts of experiences in teaching and 
administering the UICSM program. 

Newsletter No, 6, to appear early in September, is scheduled to 
contain a description of what we would suggest as a minimum course of 
study in each of the first six units if there is not time for connplete coverage. 
The units seem to expand with each Revision, but the school year does not; 
several of you have felt the sqiieeze and asked for advice on which sections 
are essential and which less essential. 

We look forward to meeting those of you who are coming to the NSF 
Summer Institute in Urbana for the first time next month, and to renewing 
friendships with those who have been here before. If you are traveling this 
summer and have an opportunity to visit us in Urbana, we will, of course, 
be very pleased to meet you, to show you how we operate, and to discuss 
with you any phase of the UICSM program. In any case, have a pleasant 
and rewarding summer. 



)v ■-,. '. 



k-i 



■■:-.] 



"i i-; ■ 



.:^i. 



.It;-. 



•ii!< ri: 



■J. :-vi • , 






TEACHING FIRST COURSE 
by Sister Mary Sarah, S. S. N. D. 

After attendance at in-service mathematics sessions in Milwaukee, 
summer institutes, mathematics lectures, and a workshop at Notre Dame 
of the Lake Training Center last August, the iinpetus came to adopt some 
project to improve Messmer High School's mathematics program. One of 
the teachers attended a demonstration course in the teaching of the UICSM 
First Course at an NSF Sumn'ier Institute at Sacramento and felt that this 
project could be successfully carried out. 

Our Principal agreed that two teachers (each with a heterogeneous 
group) naight use the course set up by the University of Illinois Committee 
on School Mathertatics. The first problem seenied to be to answer the 
student's worry as to why they were singled out. "Is it because I am a poor 
student?" or "a bright student?" It was neither reason. Each of the two 
teachers had been assigned a single algebra class and was eager to use the 
course with this one group.' Once such questions were satisfactorily answered 
there was no further obstacle. 

Students seem to have less difficulty with the UICSM course than with 
traditional algebra. Getting used to the terminology and generalizations, 
etc., poses more of a problena for the teacher, grounded in traditional 
mathematics, than it does for the students. But the Teacher's Commentary 
accompanying the text adequately provides: 

1) background information 

2) a wonderful help in solving-the exercises 

3) keys for the exercises 

4) supplementary and miscellaneous exercises 

5) tests which help determine the strengths and weaknesses of 
students in the areas taught 

Algebra taught in this way seenns to have much more meaning for the 
student. Opportunity for discovery is constant throughout the course. Exer- 
cises are well-graded, from the easy to the more difficult. Principles are 
introduced early and gradually, constantly providing a basis for work. They 
seem to tie together the ideas and to help students see continuity in their 
study. 

Interest is sustained throughout through a novel use of illustrations. 
For example: 1) the use of pronumerals, beginning with frames to generate 
other expressions; Z) the use of a diagram of a camera and a projector in 
teaching the naultiplication of real numbers; 3) the use of 'distance' and 
'direction' in taking trips to make adding positive and negative numbers nnore 



Sister Mary Sarah teaches at Messmer High School in Milwaukee, Wisconsin. 
She will attend the NSF Summer Institute for UICSM teachers in Urbana this 
summer. This article, republished here with her permission, originally 
appeared in the February, 1961, Wisconsin Teacher of Mathematics. 



•r i.f • • J- •: 



iJ' -A 



•.r!f 






:•■ ; ':iri ' . .. ^ ■ . -.y-.. 



•. ■/• 






.fi. 



--l«n< 



■ Zji,- a-. 



I i ( t •..■.•rooi'r.ur. . 

'■;'■ ■..-^'hb:- ;vu ■; 






.i.-'-'.i'i/. 



understandable; 4) using the raised ' -' sign to distinguish between the 
operation of adding a negative number and the subtraction operation. 

Traditional difficulties are obviated by the manner of teaching the use 
of grouping symbols and of preparing for teaching exponents. Even lengthy 
and seemingly involved expressions cease to look fornnidable. Sonne student 
will generally attempt unravelling the expression by applying various prin- 
ciples or the convention. The old exponent mystery has disappeared. 

With regard to initiating the course, once the UlCSM was apprised of 
our plan to teach First Course we were given every assistance. The News- 
letters, periodically sent out, provide teacher's experiences, excellent 
suggestions, and tests for the teacher's use. 

We have found teaching the course a satisfying and valuable experience 
for both students and teachers. Of course, since it is our first experience, 
our evaluation is based on the subject matter taught thus far--the first one 
and one -half units of the four required. 



ON ORDERING TEXT MATERIALS 

Schools intending to order UICSM text materials for use during the 
1961-62 school year may wish to take note of this up-to-date summary of 
their availability. 

Unit Descriptive title 

1 The arithmetic of the real numbers 

2 Generalizations and algebraic manipulation 

3 Equations and inequations 

4 Ordered pairs and graphs 

5 Relations and functions 

6 Geometry 

7 Mathematical induction 

8 Sequences 

9 Exponential and logarithmic functions 

The Student's Editions of Units 1, Z, 3, and 4 may be purchased sepa- 
rately at $1.00 per unit after July 1. The four-unit set and the Teacher's 
Edition of this set will thereafter be sold at $3. 00 and $6. 00 per copy, re- 
spectively. 

Units 5 and 6 have also been published at the following prices per copy: 

Student's Edition Teacher's Edition 

Unit 5 1. 50 3.00 

Unit 6 2. 00 4. 00 



4 - 



;.i ^.»D'..- 



■ bv 



II 



■'<■' 






The new Units 7 and 8 were described in Newsletter No. 4, pages 30-31, 
and will become available from the Press this summer. The estimated per- 
copy prices (subject to change) of these units are: 

Student's Edition Teacher's Edition 

Unit 7 1. 25 2. 75 

Unit 8 1. 75 3. 75 

Unit 9 is scheduled to appear in January of 1962; schools planning to 
use it can expect delivery near the beginning of the second semester. The 
xinit begins with rational exponents, then moves into a development of the 
thenaes suggested by the title. 

All the units mentioned above will be available from 

The University of Illinois Press 
Urbana, Illinois 

and should be ordered by unit niamber rather than course number or title, 
(There is a 10% discount on student editions when ordered in lots of 50 or 
more copies. ) 

The units on Circular Functions and Complex Numbers will be avail- 
able without charge from the UICSM Project office. Requests for copies 
should be sent to UICSM, 1208 West Springfield, Urbana, Illinois. Schools 
needing the '*old" unit on Exponents and Logarithms before the new Unit 9 
is ready should write to Max Beberman to make special arrangements. --R. S. 



'1" '1^ 



As the word "obvious,*' so also the word "proof" has a meaning which 
is dependent on the audience for whonn the proof is intended. All that is 
required of a proof is that it convince the audience of the truth of the impli- 
cation at hand. 

--R. B. Kershner & L. R. Wilcox 
The Anatomy of Mathematics, p. 77 



Whether we regard mathematics fronn the utilitarian point of view, according 
to which the pupil is to gain facility in using a powerful tool, or from the 
purely logical aspect, according to which he is to gain the power of logical 
inference, it is clear that the chief end of mathematical study must be to 
make the pupil think . If mathematical teaching fails to do this, it fails 
altogether. 

--John Wesley Young 

Lectures on Fundamental Concepts 
of Algebra and Geometry ( 1 9 1 1 ) p. 4 



■ M'' JS? 



^ li.f! 
.-■''.1 



■I •■•'••. ur: 
■' •'. '<.' t»- ■■• 



;l«.•. I 



vd • ;^-i*r-t 



>l!i 



<^-,.i.i 



* i .'.. 



rj'i ii 



;.Ji£ 



j;''h- 







i; 


•hi 

•In 


.'t 


:»H 


r 


'1- 



::f !! ,-• 

.f >H: 



1 -'••''' '^iiiilV: 



Calling All C ar tog r aph er s 

After one day spent on Glox, with a group of 13 year olds, an exercise on 
nnaps was developed. Sketches similar to the ones shown here were placed 
on the board. The students were asked: "Could this be a complete map of the 
region on Glox about which the spaceman sent messages ?" This could be 
used as homework or adapted for a written exercise in class. Adaptations 
may need to be made for older children. 

"(^ 

Long before the first space ship reached Glox, the dark side of the moon 
had been explored. Evidence had been found that intelligent beings had estab- 
lished a base there. Among the debris, papers had been discovered. Copies 
of these papers are shown on the next two pages. These pages have been 
labeled for easier identification. The drawing labeled A6 received a lot of 
attention. Everyone was certain that these were maps. But none of the 
others were recognizable. 

When the messages started coming from the spaceman on Glox, the Com- 
mander set everyone the job of studying these nnaps to see if any of them were 
maps of the regions where the spacenaan had landed on Glox. The first mes- 
sage made them discard Al as a possible map. Why? Decide which of the 
others nnight be maps for that region of Glox. Justify your decisions. 



Key 



Map 


Answer 


Al 


No 


A2 


No 


A3 


No 


A4 


No 


A5 


No 


A6 


- 


Bl 


No 


B2 


Yes 


B3 


No 


B4 


No 


B5 


Yes 


B6 


Yes 



Justification 

■ I 

Message 1 or 3 
Message 3 
Message 5 
Message 2 
Message 4 



r, Two highways 

Message 5 , . r- j t^ 

'=' between C and D 



Message 3 
Message 3 



y.v!.:. 



O L>. 



A 1 




A 2. 



A 4. 



A5. 





A6 




7 - 




8 - 



u 



.'ft 



■T- 



\ 



A Report on the Use of UICSM First Course Materials in Grades 7 and 8 

(Courtesy of Miss McCoy) 



In 1956-57, a class at the University [of Illinois] High School which was 
composed of superior 7th grade students was taught Units 1-3 of the UICSM 
First Course. The success of this experiment made it seem feasible to 
try these units with selected classes of 8th grade students in two of the 
Pilot schools. So, in 1957-58, two classes of 8th grade students at the 
Hough Street School [Harrington, Illinois] and one 8th grade class at the 
Beaver Country Day School [Chestnut Hill, Massachusetts] studied UICSM 
Units 1-3 instead of the usual eighth grade arithmetic course. The teachers, 
students, and principals of these schools were not only pleased, but enthusi- 
astic, about the results of this experimentation; as a result, these two schools 
have continued, in succeeding years, to use Units 1-3 for their superior 8th 
grade students. 

In 1958-59, 15 other schools began using UICSM First Course materials at 
the 8th grade level [in the Thomas Williams Junior High, Cheltenham, Pa. 
district, 1 class of 7th grade students was started in First Course], In these 
schools, together with Hough Street, Beaver Country Day, and University of 
Illinois High, there was a total of 27 classes studying Units 1-3 at the 8th 
[or 7th] grade level. 

In 1959-60, there were 27 of our cooperating schools using Units 1-3 for 
8th [or 7th] grade students, with a total of 45 classes. The present school 
year, 1960-61, shows a big increase in the number of our cooperating schools 
which are offering the UICSM material for 8th grade students; there are now 
51 such schools, and in these there are 95 classes. These 95 8th grade 
classes comprise approximately 31%-of the First Course classes now being 
taught in our Cooperating Schools. 

It should be pointed out that teachers using Units 1-3 at the 7th or 8th grade 
level are advised to supplennent the text, when it seems necessary, with 
work on per cents, and on intuitive geometry. --The Project staff advises 
schools to make careful selection of students for 7th or 8th grade classes in 
this accelerated program; we recommend that students chosen have IQ scores 
about 120 or above, and an arithnnetic achievement score of 8. 5 or above [the 
latter is essential, since many of the discovery exercises in Units 1-3 pre- 
suppose an ability to compute]. 

Students who study Units 1-3 in the 8th grade usually study Units 4 and 5 in 
their second year of nnathematics; they are then ready for Unit 6-Geometry- 
at the 10th grade. 

A comparison of results on our Unit tests indicates that the 8th grade students 
made scores as high, or higher, than did the students in 9th grade classes, 
A sunamary of test results is given on the next page. 



(Of 



.re.'.': (I 



r3" 



■ii 



!bt; 



or*:. • 



■•dz u 



Unit 1 

*No. of students 
Highest score 
Third quartile 
Median 
First quartile 
Lowest score 
Mean 
Standard deviation 

Unit 1 



1958- 


•59 


1959- 


60 


7th- 8th 


9th 


7th- 8th 


9th 


560 


2221 


721 


2581 


24 


24 


24 


24 


21 


20 


20 


20 


18 


18 


17 


17 


15 


14 


14 


13 


3 


1 


3 





17. 14 


16.84 


16.84 


16. 10 


4.50 


4.56 


4.34 


4.90 


1958- 


•59 


1959- 


60 


7th- 8th 


9th 


7th- 8th 


9 th 


539 


2081 


1069 


2719 


24 


24 


24 


24 


19 


18 


18 


18 


17 


16 


16 


15 


14 


12 


12 


11 


4 


2 


4 


2 



*No. of students 
Highest score 
Third quartile 
Median 
First quartile 
Lowest score 

Mean 16.24 14.97 15.1 14.4 

Standard deviation 4.30 4.57 4,4 4.5 

Unit 3 1958-59 1959-60 

•7th- 8th 9th 7th- 8th 9th 

*No. of students 271 1888 609 2571 

Highest score 24 24 24 24 

Third quartile 18 17 18 17 

Median 14 14 15 14 

First quartile 12 11 12 11 

Lowest score 4 2 3 1 

Mean 14.37 14.00 14.91 13.99 

Standard deviation 4.33 4.29 4.38 4.56 



*The number of students whose scores have been tabulated shows variation 
from unit to unit due to the fact that our office had not received test results 
from all teax:hers [or even from the same number of teachers] by the tinne 
it seemed advisable to tabulate results. 



10 - 



•. 1 



..-J. 



:-ix 






Si 



4-Vi. 
li 



•■■ . ' i 



>t. .4 



• -J ..■■L')i-'.l . 



:±l: 



NEWS AND NOTICES 




Neither Numbers nor Numerals ? 

Mr. Nathaniel Merrill, of the Newton South High School, Newton Center, 
Mass., sent in this photograph of a Volkswagen bearing 'UICSM' registration 
tags. We hope that there is some other automobile in New Hampshire inscribed 
'SMSC, and that it is not a Cadillac. 



A UICSM cooperating school, Pueblo High School in Tucson, Arizona, 
is one of three schools identified by journalist Martin Mayer as doing a 
surprisingly good job in spite of some very unfavorable socio-economic 
circunnstances. Mr. Mayer's article, 'The Good Slum Schools', is drawn 
from his recent book. The Schools, and appeared in the April, 1961, Harper's 
Magazine . 

Mr. Christ Kristo, of the Owatonna (Minnesota) High School, spoke 
before about 40 Sisters from nine elementary schools there recently in 
connection with UICSM. 



Sister Mary of the Angels, St. Rosalia High School, Pittsburgh, spoke 
to the Diocesan Board of Supervisors May 5 on the need for curriculum change 
in the Junior and senior high schools. 



- 11 



rie if 



':■■•:■):: 



!>:,(: 



:iif:-;/ ■; y. ^ v 



,j.-.; > 



>C ■.■•i<ii : 



>n(,' 















•V!-^i^'i..;|;; 






Mrs. Mary Huzzard, Cheltenham Senior High School, Wyncote, Pa., 
went to Easton, Pa., on April 1 1 to talk to their mathematics teachers 
about the UICSM materials and Cheltenham's experience with them. She 
mentions having been questioned on the extent of her students' knowledge 
of analytic geometry by a college professor who was present, and that he 
seemed not to believe her students capable of what she reported. 

Miss Olive V. Hicks, Desert Sun School, Idyllwild, California, was 
a member of the Reaction Panel at a University of Redlands conference on 
"Current Efforts to Improve Instruction in Mathematics". She took part 
in the Secondary Section meeting, devoted to "Mathematics Content Pro- 
posed for the Secondary School". 

Mr. Eugene Epperson, Talawanda High School, Oxford, Ohio, reports 
a successful open house for parents at w^hich he showed slides and talked 
about the purpose, history, and nature of the UICSM program. He also spoke 
at Miami University (Ohio) on April 27, to students and teachers in the area 
who had been invited to the campus for the day. 



UICSM Staff notes . . . Mr. Beberman took part in a mathematics 
curriculum conference at Boston University on April 26, the annual panel 
luncheon meeting of the Association of Teachers of Mathematics of New 
York City on April 29, and a secondary school mathematics workshop in 
Philadelphia on May 6; he also visited UICSM schools in New York State, 
New Jersey, Massachusetts, and Illinois early in May. . . . Miss McCoy 
visited UICSM schools in Indiana, Arkansas, Missouri, Ohio, Pennsylvania, 
and West Virginia during April and May. . . . Mr. O. Robert Brown spoke 
at the Sangamon County Teacher's Institute in Springfield, Illinois, on 
March 24. 



Everything that I say really amounts to this, that one can know a proof 
thoroughly and follow it step by step, and yet at the same time not under- 
stand what it was that was proved. 

And this in turn is connected with the fact that one can form a mathe- 
matical proposition in a grammatically correct way without understanding 
its meaning. 

Now when does one understand it?- -I believe: when one can apply it. 

--Ludwig Wittgenstein 

Remarks on the Foundations of 



Mathematics rV-25 



- 12 - 



ju; 



:T;1-./''' 



':,■ '■■■ '.n ,)-•••■ 









-7Y 



■ ;, . :iy:>X -hy^^yi 



v_yl 





ewsetter 



An occasional publication of the 
UNIVERSITY OF ILLINOIS COMMITTEE ON SCHOOL MATHEMATICS 

1208 West Springfield 
Urbana, Illinois 



Editorial 2 

How to Teach Units 
1 Through 4 in 
One Year 3 

UICSM-NETRC Tests M-P 8 

A Mathematical Description 

of Units i and 2 17 

News and Notices 26 



UICSM Newsletter No. 6 



October 24, 1961 



ioCM J 



?Jii!' 



>.■■/] '■.■■ "iH. 



8 

Vi. 



'-= ■ U S'T:: );■ ■ ■ -ivi-^' ^i' 



',C: :S:..-':ifV; »'ri-!>:;v 



!Hv;r:i>T ^f 



■r 



?::;;\TAj/;'iV!T A^ JO .'!iJ<-. 



• rtrili , :■•:;■■ 



■,f I • . 'I i> vi'.'-i 



HELLO AGAIN 

This newsletter is the sixth in a series of publications initiated by 
UICSM at the beginning of the preceding academic year. We shall continue 
to issue newsletters from time to time during 1961-62. Their prinnary 
purpose rennains the same: to provide a supplement to and amplification 
of the Teacher's Commentaries for the various text units in the UICSM 
prograna. Other items of possible interest to teachers using UICSM ma- 
terials will be included, such as test items and news of the professional 
activities of UICSM teachers and staff. 

We assume that readers of this newsletter are somewhat familiar 
with our pedagogical methods and mathematical viewpoint as exemplified 
in the UICSM text materials. It is not possible to gain such familiarity by 
a casual reading of this newsletter. [For exanriple, the third article in 
this issue contains some rather sophisticated language that is not to be 
found in the student texts. A reader not familiar with the student text 
might not realize this. ] 

Everyone who teaches or administers the UICSM program is again 
cordially requested to contribute material to our publication by submitting 
(to the Editor) anything he would like to share with others similarly occu- 
pied. We also want to hear about your professional activities in connection 
with or on behalf of UICSM, where you first encountered or have been study- 
ing the UICSM program, etc. 

Please note that this newsletter is sent only to those whom we have 
reason to think are interested in the work of UICSM, whether because they 
are teaching from our materials or otherwise. If you wish to continue re- 
ceiving it, please send to the project office your namie, address, and a 
request that you be added to or retained on our mailing list for 1961-62. 
The form on the last page of this issue may be used for this purpose, as 
well as for sending us the names and addresses of others whom you feel 
should be receiving this publication. --Ed. 



IN ONE YEAR 



We are ready to admit that the complete four-year UICSM program will 
take more than four years tc teach if all teaching is done in the recommended 
UICSM "style". If a school begins the UICSM program in grade 9 for unse- 
lected college preparatory students and wishes to take at least some of these 
students through the four- year program to the point where they will be able to 
pursue a college freshman course in analytic geometry and calculus, it will 
be necessary to de-emphasize certain topics and to speed up the presentation 
of some sections. Here- are sonae suggestions for time allocations and teach- 
ing procedures for Units 1-4 that will enable a class to cover this material in 
one school year. 

Unit 1: The Arithmetic of the E.eal Numbers 

The nature of this unit is such that no topic can safely be omitted. Each 
topic is important either for its own sake or because it provides inriportant 
ground work for topics in later units. 

The number -numeral problem should take one class lesson plus one 
assignment, the assignment bsing, parts C^, D, and ^ on pages 1-N and l-O, 
Discussion of the Stan-Al corrsspondence should lead quickly to the point that 
in teaching somebody mathematics, you must acquaint him with mathennatical 
entities such as numbers. In order to talk about these entities, you need names 
for them--for example, nunnerals. So, another part of teaching mathennatics 
is acquainting learners with names. In order to talk about these names, you 
need names for them - "for exa,mple, names of nunrierals. It is for this purpose-- 
forming names of nannes- -that single quotation marks are used. 

Take at nnost 3 lesions for pages f- 1 through 1-15. You should be finished 
with page 1-23 at the end of the fifth lesson. 

One secret in making rapid progress is to devote a good chunk of the class 
hour to "supervised study". A.n i'ncredible amount of class time is spent in 
compelling students to listen to explanations and to participate in redevelop- 
ments of topics required bv only a few students. 

One lesson plus one assit5;nment will take care of pages 1-24 through 1-28. 
Let students read pages 1-29 through 1-32 at their seats, you summarize the 
main point for thenn, zip through parts A a.nd ^ on page 1-32, have five minutes 
of fun in class with part C^ on page .1-33, present the problem of ambiguity as 
discussed in the lower third of page 1-34, and use supervised study for students 
to read pages 1-33 through 1-35, plus homework on page 1-36. 

Another notorious time -consumer is the checking of homework in class 
by reading answers and discussing points of disagreement. Why not duplicate 
a list of answers and pass this out either when checking time comes or, per- 
haps, when the assignment is niade ? For example, you might let students 
get started on part A on page 1-35 and pass out the answer key after three 
minutes of work. Of course, you would probably omit the "answer" to exer~ 
cise 34. (You might even prefei to give the answers for just the even-numbered 
exercises even though you assigned all of them. ) 

-3- 



Pages 1-37 through 1-42 are worth two lessons. And so are pages 1-42 
through 1-49. One lesson can take care of 1-50 through 1-54 plus homework: 
part A on 1-55 and Supplementary Exercises, part _G. 

Pages 1-56 through 1-59 are critical. Spend two or three lessons on 
this with much class discussion. 

Pages 1-60 through 1-66 can be handled in one supervised study lesson 
plus one honnework assignment. 

Pages 1-66 through 1-72 are critical. Spend two lessons on this. 

Three lessons including a lot of supervised study will take care of pages 
1-73 through 1-79. 

Three more lessons on 1-80 through 1-85 with a lot of class discussion 
on 1-83 and 1-84. 

One solid supervised study lesson plus homework will take care of 1-86 
through 1-92. Another day will bring you through the middle of page 1-95. 

Take another day for mixed dtill work on operations. 

Three lessons are necessary for the important work on pages 1-95 through 
1-102. 

Two more lessons will do it for 1-103 through 1-110. 

The above schedule calls, for some 29 lessons. If you throw in tinne for 
tests and review work (but not too many assemblies), you can complete Unit 1 
in 7 weeks. 



Unit 2: Generalizations and Algebraic Manipulation 

As noted in the introduction to the commentary for Unit 2, the purpose 
of this unit is twofold: proving theorems, and applying these theorems to 
the manipulation of algebraic expressions. Just as it is unreasonable to 
expect complete nnastery of manipulation in the space of one unit, so is it 
unreasonable to expect students to write the proofs of all the theorems in 
the unit. The important thing is that each student come to realize that any 
manipxilation maneuver can be justified deductively- -that it is a consequence 
of the basic principles of the arithmetic of the real numbers. 

Pages 2-A through 2-1: one lesson plus one homework assignment, w^ith 
10 minutes of a second lesson on pages 2-G through 2-1. 

Pages 2-1 through 2-7: one supervised study lesson plus one assign- 
ment. A bit of discussion on exercise 2, page 2-3; No. 6 on 2-4; No. 3 on 
2-6. 

Pages 2-8 through 2-13: two lessons plus two assignments. 



Pages 2- 14 th::ough 2-23: three lessons plus three assignments. Read 
2-14 through 2-16 in class- Let students read 2-17 and 2-18 by themselves. 
Supervise their work on page Z-19. At least half of the work on page 2-21 
should be done under your supervision. This material is crucial. 

Pages 2-23 through 2-27: one lesson plus one assignment. This is ex- 
tremely impoi'tant material, especially page 2-27. 

Pages 2-28 and 2-29: optional. 

Pages 2-30 through 2-3S: three lessons plus homework. Critical material, 
calling for masterful teaching. This is the point at which students get a chance 
to see the purpose of procf and to use free variables in test-patterns. 

Pages 2-39 through 2 "44: one lesson plus homework. 

Pages 2-45 through Z-59i four lessons plus homework, including super- 
vised study. 

Pages 2-60 and 2-6 'i: one lesson plus homework (which should be care- 
fully checked). 

Pages 2-6?, through 2-66: two lessons plus homework. Parts A, B, and 
C^ can be done orally in class. Much careful teaching must be done with 
pages 2-64 through 2"o6, for it is here that students get their first experience 
in proving conditionals. 

Pages 2-67 through 2-/6: three lessons plus homework, including much 
supervised study over pc.ges 2-71 through 2-75. Students shoiild know what 
the theorems on. pagec 2-69 and 2-70 are about. Each student should be able 
to prove at ] jast cue theorem from part A and one theorem from part B. 

Pages 2-77 through 2-00 (including the relevant Supplementary Exercises): 
two supervised study periods plus homework. Some of this could be blended 
with the v/ork on pr.g?t; 2-72 through 2-74. 

Pages 2-81 throti&h 2-84: one lesson plus homework from the Miscel- 
laneous Exercisos or Supplementary Exercises. 

Pages 2-85 thron^-h 2-91: three lessons plus homework. 

Pages 2-92 throv.^h 2-108: eight lessons plus homework, including much 
supervised study on drill lessons. Proofs of theorems need not be assigned 
as homework except as optional work, but some proofs should be presented at 
the bl:ickboai-d. 

Pa.ges 2-109 th.ro^^^,h '.;-l''. 1: cr.e Issson plus homework. 

Pages 2-112 through 2-153; six lossons. Some of this work should be 
distributed over the entire unit. 

This schedule calls loz- some 40 lessons. 



I'. 



-5- 

I 



Unit 3: Equations and Inequations 

There are blocks of material in this iinit that shoiild not be taught as units, 
but should have other materials interspersed within them. This is the case 
for the work on pages 3--40 through 3-50, 3-58 through 3-82, and 3-174 through 
3-185. Such programing is not indicated in the schedule suggested below. 

This unit offers many opportunities for supervised study, and many oppor- 
tunities for time-wasting activities. Unless you consciously try to avoid it, 
you will find yourself trapped into forcing the entire class to listen to your 
explanation of a.n equation or a word problem which troubled only a few stud- 
ents. 

Pages 3-A through 3-3: one lesson. 

Page 3-4: option?.L 

Pages 3-5 through 3-10.. tv/o lessons. 

Pages 3-11 through 3-18: three lessons with much supervised study. 

Pages 3-19 through 3-Zl:, one lesson. 

Pages 3-22 through 3-25: one lesson. 

Pages 3-26 through 3-32: three lessons with much class discussion. 

Pages 3-32 through 3-5C: five lessons with much supervised study. 

Pages 3-51 through 3 -57: three lessons including Supplementary Exer- 
cises on page 3-173, 

Pages 3-58 through 3-32; eight lessons including Supplementary Exer- 
cises on pages 3-17i through 3-185. 

Pages 3-83 through 3-95: four lessons^ 

Pages 3-96 through 3-99: two lessons. 

Pages 3-100 through 3-107: four lessons. 

Pages 3-108 through 3-111: two lessons. 

Pages 3-112 through 3-120: two lessons, mostly supervised study. 

Pages 3-121 through 3-123: one lesson plus homework, mostly super- 
vised study. 

Pages 3- 127 through 3-130: one lesson. 

Pages 3-131 through 3-133: one lesson. 

Pages 3-134 through 3-136; tv/o lessons. 

Pages 3-137 through 3-197;: eight lessons. 

This schedule calls for some 54 lessons. 

-6- 



"'■.••:jf--i '1- 



«.':j- 



. v'-.K? ; :'■.'■■■ 



Unit 4: Ordered Pairs and Graphs 

This unit has many potentialities for enrichment. It is particularly dis- 
tressing to pare it down to a minimum course, but. . . 

Pages 4-A through 4-H: three lessons, one of which should be spent on 
parts D and E. 

Pages 4-1 through 4-4: one lesson. 

Pages 4-4 through 4-10: two lessons. 

Pages 4-11 through 4-19: three lessons. 

Pages 4-20 through 4-25: one lesson [part B^ optional]. 

Pages 4-26 through 4-35: seven lessons. 

Pages 4-36 through 4-41: one lesson. 

Pages 4-42 through 4-47: two lessons. 

Pages 4-49 through 4-51: one lesson. 

Pages 4-52 through 4-55: one lesson. 

Pages 4-56 through 4-58: two lessons. 

Pages 4-59 through 4-64: three lessons, including Supplementary Exer- 
cises. 

Pages 4-65 through 4-70: three lessons. 

Pages 4-71 through 4-86: seven lessons. 

Pages 4-87 through 4-90: one lesson [exercises 11 and 12 optional]. 

Pages 4-91 through 4-95: three lessons. 

Pages 4-95 through 4- 13 1: six lessons. 

This schedule calls for some 47 lessons. 



So, by following the schedule suggested above as closely as possible, 
you should be able to cover all four units in about 170 lessons. This does 
not include time for testing, assemblies, field trips, etc. Class time can 
be conserved by making some tests of the take-home variety. And, in order 
to gain more time, you might consider forming a faculty committee on de- 
creasing the number of assemblies! -- M. B. 

v<, vl, -.1^ 

'I- 'I- '1^ 



-7- 



.1-JL,- -:£•>; 



^• uiijc[> 



:-i5 ! 



s}'u- '■•.'-'• 



UICSM-NETRC Math Study Tests 

Four more Tests from the film study series are reprinted below in 
their original versions. Those designated A, B, C, and D were in News- 
letter No. 1, E through H in No. 2, and I through L in No. 4. The page of 
the text to have been completed before the test is given is indicated in 
brackets. Hopefully these items will be of sonme interest and value to 
teachers when they begin teaching Unit 3. --R. S. 

Test M [3-18] 

Matching. Remember : an answer may be used once, or twice, or not at all. 

Use these choices for questions 1 through 4: 

(A) (x: XX > 1} (B) {x: xx = x} (C) {x: xx - 1 = O} 

(D) {x: |x| < 1} (E) none of these 



1. -1, 1 = 2. 1, -1 = 3. {0, 1} = 4. {-1, 1} = 

Use these choices for questions 5 through 8: 

(A) (B) {5} (C) {5. -5} 

(D) {x: X + 5 = X + 5} (E) none of these 

5. {x: |x| = 5} = 6. (x: XX + 5 > 0} = 

7. (t: t(-t) = -25} = 8. (x: X + 5 = x} = 

Use these choices for questions 9 through 12: 



(A) -1, 1 (B) -1. 1 (C) -1. 1 



-> 



(D) -1, 1 (E) none of these 

9. {x: |x| < 1} = 10. {x: -1< x< 1} = 

11. {x: X > 1 and x<-l}= 12. (x: xx< lorx = -l} = 



Choose the one correct answer. 

13. The midpoint of a number line segment is 2 and one end point is 10. What 
is the other end point ? 

(A) -6 (B) 6 (C) 8 (D) 18 (E) none of these 



I ... f •- •, 



:C/ ( 



i ^ ' •:. r,i^ 



14. The end points of one number line segment are -2 and 6. The end points of 
another are 4 and 16. The midpoint of the segment joining their midpoints is 

(A) 4 (B) 5 (C) 6 (D) 7 (E) none of these 

15. For each x, if the end points of a segment are (4x + 7) and (2x - 3), the 
midpoint is 

(A) 2x + 10 (B) 3x + 2 (C) 6x + 4 (D) 8x - 21 (E) none of these 

16. Which of these sets contains no elements ? 

(A) {a: -aa > 0} (B) {0} (C) {b: 7b + 5b = lib} 

(D) {0} (E) none of these 

17. Look at these names of number line loci: 

{x: X > 1} {y: |y| = l} {s: ss > O} 



{m: m < or m > 1} 0, 1 1, 

There are, among the loci named, exactly 

(A) no intervals and no segments 

(B) no intervals and one segment 

(C) one interval and no segments 

(D) one interval and one segnaent 
. (E) none of these 

18. Look at these names of number liJie loci: 



1, (z: -1 < z < 0} {q: |q| = l} 



{d: -d< 0} ■ 1, 1 {0} 

There are, among the loci named, exactly 

(A) no half-lines and no rays 

(B) no half -lines and one ray 

(C) one half -line and no rays 

(D) one half-line and one ray 

(E) none of these 

19. Which of the following is a name for this number line graph? 

^ 1 ^ 

1 2 



(A) 0, 2 (B) 0, 2 (C) {x: [x] < 2} 

(D) {x: x> and x < 2} (E) none of these 

-9- 



ic aJ/i I •■•• 1'-': 



.rt: .(; - y'::'i hr^ <* •;• /;•; ■'•'£• J/n 



.•*'/ 



■'V :d 



.}. 



.^■■f. »•..; Or 
...■,.•<>>• .•■•'ft >'-lt-i' 



IT 



rl,: :;<. 



20, Which of the following is a name for this number line graph? 

3 h— 

-1 1 

(A) {w: w^v< 1} (B) (x: |x|< l} (C) {f: -1< f< l} 



(D) -1, 1 (E) none of these 

Key for Test M [3-18]: 



1. 


D 


2, 


E 


3. 


B 


4. 


C 


5. 


C 


6. 


D 


7. 


C 


8. 


A 


9. 


E 


10. 


B 


11. 


E 


12. 


B 


13. 


A 


14. 


C 


15. 


B 


16. 


A 


17. 


A 


18. 


E 


19. 


D 


20. 


C 







Test N [3-31] 

In each of problems 1 through 10,, two sets P and Q are described. 
Mark A if P C Q and Q C P, 

B if P C Q and Q ^ P, 

C if Q C P and P ^ Q> 

D if P ^ Q and Q ^ P. 

1. P = {x: |x| =1}; Q = {x: xx = l} 

2. P = {x: |x - 4| = 1}; Q = {x: xx + 3 _= 4x} 

3. P = {x: XX = x}; Q = {s: 3s + 4s = 5s} 

4. P = {y: i5y - 7 = 5 + 9y}; Q = {O, 2} 

5. P = {£: 5f + 3 - f = 13f - 9f + 3}; Q = {g; 2g = 3{5g - 4g)} 
5= P = {m: 7m - 3 = 89}; Q = (m: (7m - 3) + 3 = 92} 

7. P = {n: 5(2n - 3} = {17 - n)5}; Q = {n: 2n - 3 = 17 - n} 

8. P = {x: (x + 7)x = 5x}; Q = {y: y + 7 = 5} 

9. P = {z.: 81 - 3z = 78}: Q = {v: 81 = 78 + 3v} 

10. P = {x: (:: H- 7)(x - 5) = (x + 7)(x -;- 2)}; Q = (x: 7(x - 5) = 7{x + 2)} 

Which is the correct solution set, if listed, for these equations? 

11. X + (x + 2) + (x + 4) - 27 

(A) (B) {7} (C) {21} (D) none of these 

-10- 



r 



12. |3 - 2m| = 1 

(A) {1} (B) {2} (C) {1, 2} (D) none of these 

13. XX + 1 = 

(A) (B) {1} (C) {-1} (D) none of these 

14. 2=3- 9yy 

(A) (B) {^} (C) {^} (D) none of these 

15. 7(5z - 15) + 4(92 + 12) = 25 

(A) (5} (B) {6} (C) {7} (D) none of these 

16. 6x - 5 + 3x{2 - x) - 4x(3 - x) = 116 

(A) {-11} (B) {-11, 11} (C) {121} (D) none of these 



III. Choose the one correct answer. 

17. P and Q are sets. If P is a subset of Q and 3 is not a mennber of Q, 
what follows ? 

(A) {3} is a subset of P 

(B) there is a member of Q which is not in P 

(C) 3 is not a member of P 

(D) none of these 

18. How many subsets of {l, 2} are there? 

(A) 4 (B) 3 (C) 2 (D) none of these 

19. What is needed to show that {x: 3x + 5 = 17} C {x: 3x = 12} ? 

(A) V V V if X = y then x + z = y + z 

X y z ■' ' 

(B) VVV ifx + z=y + z then x = y 

X y z ■' ■' 

(C) both (A) and (B) 

(D) none of these 



-11- 



.C \(: : .cJ: 



It.: I.,.- 



20. Why is (y / 0: — + 3 = 8} = {y / 0: (— + 3)y = 8y} ? 

(A) VVV /-ifx = y then xz = yz 

X y z ;= ■' ' 

(B) V^V^V^ ^ ^ if xz = yz then x = y 

(C) both (A) and (B) 

(D) none of these 



Key for Test N [3-31]: 

1. A 2. D 3. C 

8. C 9. A 10. C 

15. B 16. B 17. C 

Test O [3-57] 

I. For each pair of equations below, choose the most immediate justification 
for the step from the first equation to the second. Mark 

A if the justification is 'equivalent expressions', 

B if the justification is 'the addition transformation principle', 

C if the justification is 'the mvilti plication transformation principle', or 

D if none of these justifications apply. 

1. 5x - 13 = 8 - 7x^ 2. T - 7 =2 + 4 



4. 


B 


5. 


C 


6. 


A 


7. 


A 


1. 


B 


12. 


C 


13. 


A 


14. 


D 


8. 


A 


19. 


B 


20. 


C 







5x - 7x - 13=8 



3 4 



3. 5y - 3 = 9 + 2y | 4. 

5y- 3 + 3 = 9 + 2y+3J f"\ / , " _ ^ ^ > [x /^ 0] 




^ + 



y + 3 y + 3 



[y/-3] 



IL Choose the correct solution set for each of the following, or mark (D) 
if none are correct. 

6. XX = 3x 

(A) (B) {3} (C) {0, 3} (D) none of these 

-12- 



(tl) 



7. |+5=x + ^ 
(A) {1} 

12 3 



3x - 7 7 - X 
(A) {5} 

4 8 



X + 6 X - 6 
(A) {-3} 



(B) {j} (C) {-2} (D) none of these 



(B) {7} (C) Iff? (D) none of these 



(B) {6} (C) {18} (D) none of these 



10. 5{x - 7) = (4 - 3x){7 - x) 

(A) (B) {3} (C) {7} (D) none of these 



II. Choose the correct result when each of the following equations is solved 
for 'x*. 



11. ax + by + c = 



(A) x = ^^. [a,^0] 
(C) x = -^^^. [a;^0] 



by - c 
a 



(B) x = "^^ ^ , [a 7^0] 



(D) none of these 



12. a + bx = x + c 

c - a 
1 - b 

a + bx 



(A) X = 
(C) x = 



. [c/0] 



(B) x = -^-:^. [b/1] 



(D) none of these 



13. 1+1=5 
x y 



(A) X 



_ 1 - 5v 



I 



(c) >^ = T^' ^y^h 



14. y = — - n 
' X 



(A) x = ^. [ny/0] 



<^) ^ = ^' fy^-'^i 



(B) X = y(5x - 1) 



(D) none of these 



(B) x = 1-^-2^, [y/0] 



(D) none of these 



13- 



• i^•■"v^( 



:^> t' '' 



: 5 



•V) 



15. A = 



Bx 

C + Ex 



(A) x = 
(C) x = 



AC 



, [AE ^ B] 
AC - B ' t^^ ^ ^i 



AE - B 
AC 



(B) X = ^^ae"^^ . [AE;^0] 



(D) none of these 



IV. Choose the correct solution set for each equation from among these: 

(A) (B) {0} (C) the set of all real numbers 

Mark (D) if none of these is correct. 

16. X + 3 = X - 5 

17. 3x = 5x 

18. 3 + 4 = 5 

19. (x + 3)(x - 3) = XX - 9 

20. 2+2=4 



Key for Test O [3-57]: 

1. D 2. C 3. B 

8. A 9. D 10. D 

.15^ D 16. A 17. B 



4. 


C 


5. 


C 


6. 


C 


7. 


C 


11. 


C 


12. 


B 


13. 


D 


14. 


C 


18. 


A 


19. 


C 


20. 


C 







Test P [3-82] 

Choose the correct ajiswer to each of the following problems. If the correct 
answer is not given, mark (D). 

1. For each number x of arithmetic, x gallons of a 25% alcohol solution 
contain ? gallons of alcohol. 



(A) f 



(B) 25x 



(C) 



X 



25 



(D) none of these 



2. A dog chasing a rabbit, which has a start of 150 feet, jumps 9 feet every 
time the rabbit jumps 7. The dog will overtake the rabbit in ? leaps. 



(A) 300 



(B) 150 



(C) 75 



(D) none of these 



3. A student has quiz scores of 100 and 84. What score must he achieve on 
a third quiz in order to have an average score of 90? 

(A) 86 (B) 94 (C) 96 (D) none of these 



14- 



J,- 



?•! id'::. ; IS. 



~.;.- 7 x .: > 



. ■. i 



-/;>yf 



{$.■■',■' ' " V^-jT 



_0 i I jJj-". 



•*.5. y^..' .rr:.i 
■ "-if ■; ' ■ . • '!: 



!t ioo 



4. For each number J of arithmetic, for each aon^ero numoer K of arithmetic, 

J is ?_ times hs la.rf:a as Ko 

(A) JK (B) ~r (C) — (D) none of these 

5. For each u, for each v, wna!; must be Fubtricted from u to yield a differ- 
ence of V ? 

(A) u - V (B) V + u (C) V - u (D) lione of these 

60 For each number E of a:.-:thmctiCj for each non!<:ero number F of arithmetic, 
E is ? "/j of Fo 

(A) ^^-^ (B) -p- (C) -■—'- (D) none of these 

7. For each :iujTibc:: .1 i of arithmcti',, for each uumher K of arithmetic, if a 
train runs M miles in 5 ruur.:.- > ov/ nm.ny mi^ec will it run in K hours at 
the same r-'tc o-.' np,2£d? 

(Ai ^■';- (3) ■■■|"- ;C*, 5I-.I.K (D) none of these 

8. Charlie Brown is Lh.'n'ing of a aumb-^r,, Ji he multiplies it by 4 and then 
adds 5 to the prodi-.c^c, he [',ecG a si-.n^ of 12, V/hat nu^nber is he thinking of? 

(A) 4 (B) 7 (C) Z (D) none of these 

9. Delores Haze ifj thinking, of ;j. iji^T'bf::-, If rhe subtracts 251 from it and 
then divide? the c.ifjorcnr.c ly 'L .^'.le g''--"" '■• ^.'-lo^'i-rj-t of 13o What number 
is she thinking of? 

(A) 3-:-2 :.B) 2j.?.-~ (C) ~ 16 J (D) none of these 

10. If you incvease k c.ji-t.-.i:.. raimboi by 1 '/, you get '^he ?ame result as if you 
had subtracted b?Jf I'hs i.u..;nbv-.r ^- cm 5» "'Vnat is this number? 

(A) -6 (o) --.: :C/ (D) none of these 

11. K a house and. a lo<- zo:^s'i\ic.T r.ojt $23, 000 -nd the house costs $20, 000 
more than the Ic.'., ..h'! lo': r.lo.uc ■. jc^to $ ? . 

(A) 2, QC'O (E) JjSOO (C) 1, OGO (B; rone of these 

12. In a certain grcv.p of 6OC jj ;;:■'. .- 'i^.r :'i"i:.':S.^s of them ere men. Four- 
fifths of the people in <:he j^ro-..-V) i'.:.-e murried. There are ? married 
men in the group, 

(A) 360 *E.' 240 (C; 233 p} none of these 



■..5- 



•I ■ . ir.i;;: 



13. A rectangular flower garden is 20 feet longer than it is wide. Its peri- 
meter is 360 feet. The length of the flower garden must then be ? feet, 

(A) 160 (B) 80 (C) 100 (D) none of these 

14. The quarterback of the Zabranchburg High football team has completed 7 
out of 14 passes in a practice session. The coach says the squad will keep 
practicing until 65% of the passes are complete. The quarterback must 
throw at least ? more passes before practice is over. 

(A) 8 (B) 6 (C) 14 (D) none of these 

15. A hiker can average 2 miles per hour going uphill and 6 miles per hour 
gcing downhilL His average speed for a trip to the top of the hill and back, 
if he spends no tinne at the top; is ? miles per hour. 

(A) 3 (B) 4 (C) 8 (D) none of these 

16. For each x, if x is a number of arithmetic between and 50, a nnan 
traveling 100 miles at x miles per hour a.rrives at his destination 2 hours 
late. How many miles per hour should he have traveled in order to 
arrive on time ? 

/A\ 50 - X ,_ , 5 Ox ,„, 10 - 2x ,^, , . , 

(A) — tt; (B) -z. (C) (D) none of these 

5 Ox 50 - x X 

17. A small pump can fill a pool in 6 hours. A larger pump can do it in 2 
hourSo Two small pumps and one larger one, working together, can fill 
the pool in ? hour(s). 

(A) 4 (B) 1 (C) 1.2 (D) none of these 

18. K the length of a rectangle is increased by 20% and its width increased 
by 10%, the area of the rectangle will be increased by ? %. 

(A) 15 (E) ZO (C) 30 (D) none of these 



Key for Test P [3-82]: 

1. A 2. C 3, A 

8. D 9. A 10. D 

15. A 16. B 17. C 



4= 




5, 


A 


6. 


A 


7. 


B 


1. 


c 


12, 


C 


13. 


C 


14. 


B 


8. 


D. 















-16- 



■ fHi:!'. 



A MATHEMATICAL DESCRIPTION 
OF LNIT3 1 AND 2 



As a starting point, we assume that the students are somewhat familiar 
w^ith what we call the system of nunnbers of arithnaetic . This system can be 
back-handedly described as the system of "unsigned" real numbers. It is 
isomorphic --with respect to ordering and the fundamental operations- -to the 
system of nonnegative real numbers. The real numbers can be defined to be 
equivalence classes of ordered pairs of numbers of arithmetic. This is not 
done in the text; the possibility is mentioned in the COMMENTARY for 
page 1-1 of Unit 1, TC[l-l]a and b. Thus, in terms of concepts with which 
students become acquainted in Unit 5 (Introduction and 5.01), each real 
number is a relation among the numbers of arithmetic; for example, *Z is 
the relation of being 2 greater than, and "nTS is the relation of being \f3 less 
than. 

Although the authors may think of a number of arithmetic as a set- 
theoretic entity of a certain kind--a set of sets of ordered pairs of finite 
cardinal numbers--and of a real nuinber as a set of ordered pairs of num- 
bers of arithmetic, the familiarity with the numibers of arithmetic which 
may be expected of students has, of course, a different basis. A student's 
feeling of being acquainted with the numbers of arithmetic probably has its 
origin in his experiences of using numerals for these numbers to formulate 
accepted answers to questions concerning measures of magnitudes such as 
lengths, areas, weights, etc., and of manipulating such sumbols to obtain 
accepted answers to further questions of this kind. That these numerals, 
since they function as nouns, refer to entities of some kind, is an inference 
which, justifiably or not, he makes without much conscious thought. For 
him, numbers of arithmetic are things whose names occur in a character- 
istic way in sentences about measures of magnitudes. 

This attitude towa.rd the numbers of arithmetic makes a good starting 
point from which to develop a similar feeling for the real numbers. [See 
pages 1-1 through 1-4 and TC[l-2, 3]..] For exam^ple, this morning the 
dollar-measure of the money in my pocket was 3. 59, and it is now 3. 22. 
By a physical process analogous to subtraction the magnitude of my cash- 
on-hand has decreased by a magnitude whose dollar-measure is 0. 37. 
Speaking somewhat loosely, the magnitude of my solvency has undergone a 
change which involves both a magnitude and a direction. This change can 
be described , as above, by giving its direction and a measure of its magni- 
tude. Can it, itself, be measured ? It seems reasonable that changes in 
naagnitude should be measurable, but that their measures will be numbers 
of some kind other than the numbers of arithmetic, which are measures of 
magnitudes. That there are such other numbers--the real numbers--be- 
comes an article of faith, en the same level as the student's belief in the 
existence of the numbers of arithmetic. In particvilar, the dollar-measure 
of the change in the magnitude of my cash-on-hand is the real number "0. 32. 

It is to be noted that, on its own level, the above method of calling 
attention to the real number, s parallels the author's underlying conception 
of the real numbers as relations among the numbers of arithmetic. Magni- 
tudes are measured by numbers of arithmetic^ and changes in magnitude-- 
which can be identified with relations among inagnitudes (for example, the 
relation of being 0. 37 dollars less "han/'-are measured by relations among 
the numbers of arithmetic. 

-17- 



Throughout the course students are led to build new mathematical 
concepts from earlier ones by ways which, like the foregoing, parallel the 
relationships which the former have to the latter in the underlying mathe- 
matical philosophy adopted by UICSM. As a result of such an approach, 
students see the parts of mathematics they study as parts of a single subject 
rather than as a collection of somewhat disparate topics. Consequently they 
have a better understanding of what matheinatics is, and develop more power 
and freedom to make use of what they learn. 

Returning to the students' observations of real numbers, it is now easy 
to discover how to connpute the measure of the change in a magnitude which 
is the result of two successive changes whose measures are known. [See 
pages 1-5 through 1-8 and TC[l-8]a, b. ] This discovery focuses attention 
on a certain operation on real numbers and motivates practice in finding the 
result of applying this operation to given real numbers. Thus students first 
discover the operation of addition of real nunnbers and become convinced of its 
utility before deciding that it is reasonable to call this operation 'addition'. 
Among the advantages of this approach is the by-passing of certain difficul- 
ties which may arise when addition of real numbers is introduced by a 
definition which, in part, tells the student that in order to find the sum of a 
positive and a negative number he should begin by finding the difference of 
the two corresponding numbers of arithmetic. 

A similar device, dealing with rates of change, draws the students' 
attention to another operation- -multiplication of real numbers--and stimu- 
lates him to practice it. [See pages 1-17 through 1-22.] The procedures 
which students develop for computing sums and products of real numbers 
lead easily to an awareness of the isomorphism--as far as concerns addi- 
tion and multiplication- -between the system of the numbers of arithmetic 
and that of the nonnegative real numbers. Quoting the text [page 1-30J; 

the nonnegative r£al numbers act 
like the numbers of arithmetic with 
respect to both addition and multi- 
plication. 

The isonnorphism just referred to suggests that it will often not be 
confusing to use the sanne symbol (for example, '3') either as a name for a 
number of arithmetic or as a name for the corresponding nonnegative real 
number (i.e., either as a name for the number 3 of arithmetic or as a 
name for the real number "^3). 

Students' previous experience with the operations of addition and mul- 
tiplication of numbers of arithmetic makes it easy for them to become aware 
of the commutative, associative, and distributive properties of these opera- 
tions, and of the neutral character of the numbers and 1. [See pages 1-44 
through 1-48.] They see the value of being aware of these properties by 
discovering that they can be used to predict the successful outcome of a 
shortcut. [Page 1-56] This is also preparation for discovering proofs. The 
isomorphism previously noted suggests that addition and multiplication of 
real numbers may have similar commutative, associative, and distributive 
properties, a suggestion which is strengthened by carrying out some com- 
putations. [Pages 1-60, 61 and TC[l-60, 6l]a, b] Such testing by compu- 
tation prepares students to recognize, accept, and use instances of, for 



18- 



/C.I • j<--,:r 



• ■J i 






;!■;':» : ori ■ 



■ i>v.•.^•^^J,^■ 



'I- 1 m ,. . 



■i ;l:i. 



•jj/ft -J I- s 



:U> ■."':r.V . 



■.{\:, -.,1^ . ■ ir.:; •. ua; 



example, the associative priaciple for multiplication, such as: 

*385. 7M"56.Z' |) = (*385.7«-56.2)« | 

without testing them. 

At this point (in Unit 1) students are still working with only the arith- 
metic of the real numbers. They have not yet been introduced to variables 
and so have no efficient way of stating the principles, for example: 

V V V x(yz) = (xy)z, 
X y 2 ' ^ / / ' 

whose instances they now recognize and accept. This introduction comes 
early in their study of Unit 2, during the second quarter of their first UICSM 
year. By this time they have gained s\ifficient familiarity with instances of 
the principles to have little trouble in formulating the principles for them- 
selves once the appropriate linguistic devices of variables and quantifiers 
have been introduced. 

In the remainder of Unit 1 students use instances of these "basic 
principles for real numbers" at appropriate points to justify connputational 
short cuts. This is valuable preparation for learning, as they do in Unit 2, 
to derive theorenns from the basic principles. The interplay between the 
process of discovering real number properties through computation, and of 
formulating such discoveries and deriving the formulations from the basic 
principles, has much to do with the growth of the students' understanding 
of mathematics and of what mathematics is. 

UICSM students are unlikely (to say the least) to adopt the usual lay 
viewpoint that the world's greatest mathematician is the idiot -savant who 
can perform the nnost astonishing feat of mental computation. The value of 
the interplay between discovery and proof is underlined by the first part of 
Hadamard's statennent: "The object of mathematical rigor is to sanction 
and legitimatize the conquests of intuition, and there never was any other 
object for it. " 

Students are next led to conceive of a singulary operation, such as 
that of adding "5 or of multiplying by *Z, as a set of ordered pairs. [Pages 
1-67, 68, 75, 76] For example, one mennber of adding ~5 is the pair {"^7, "^2). 
Defining subtraction as the inverse of addition- -in particular, subtracting "5 
is the operation that undoes what adding "5 does--leads rapidly to recog- 
nition of the fact that the pairs belonging to a subtracting operation are the 
converses of those which belong to the corresponding adding operation. 
Example: because (*7, *Z) belongs to adding ~5, ("''2, """7) belongs to subtract- 
ing "5, Moreover, it turns out, on comparison of their members, that the 
operation of subtracting ~5 is the same as the operation of adding *5. In 
general, subtracting a real number is the same as adding its opposite. (In 
Unit 2, this is formulated as the principle for subtraction : 

VV x-y = x+— y. 

x y ■' ■' 

Together with the principle of opposites : 

V X + -X = 

X 

and the previously mentioned principles concerning addition and multiplication 
it furnishes a basis from which to derive theorems about subtraction. ) 



19- 



. , .-' n 



,■ r'iv 



V.., a J ■•'-'. t.-.ijr: ,..S 



Uifi''*' 



.b. : 



I':: ■■■■^ 



■>1^ • iiii^:^^*=^'' 



Dividing by a nonzero real number is introduced as the inverse of 
multiplying by that nunaber, and an exercise suggests that multiplying by 
has no inverse. The further development of this insight is reserved for 
Unit 2. 

The introduction of the basic concepts involved in the structure of the 
real number system as an ordered field is completed by defining the relation 
less-than. After doing this there is some practice with the notations '<', 
*>', 'f^', '>', etc. Solution of inequations is taken up in Unit 3 and the de- 
ductive organization of the theory of order is undertaken in Unit 7. However, 
in Unit 2, the use of '< ', etc., significantly increases the variety of exer- 
cises which can be fornnulated. It also gives students an opportunity to 
become accustomed to considering inequations to be as "natural" as equa- 
tions are. 

The final topic in Unit 1 is the absolute value operation. For peda- 
gogical reasons this operation is defined, here, as an operation from the 
real numbers to the numbers of arithmetic. Its converse is the union of 
the two operations "positiving" and "negativing", which map the numbers 
of arithmetic on the "corresponding" nonnegative and nonpositive numbers, 
respectively. The three operations--absolute valuing, positiving, and 
negativing- -make it possible, in Unit 2, to formulate definitions of addition 
and multiplication of real numbers in terms of the corresponding operations 
on numbers of arithmetic and, in general, to pass readily back and forth 
from one system to the other. In later units we introduce the more usual 
use of 'absolute value' to denote a mapping of the real numbers on the non- 
negative real numbers. This second mapping is, of course, the composi- 
tion of the positiving operation with the original absolute value operation. 



The purpose of Unit 2 is twofold: to help students become proficient 
in the elementary techniques of symbol-pushing used in simplifying algebraic 
expressions, and to lead them to discover, prove, and use the theorems 
about numbers which justify thes.e techniques. The attainment of the first 
of these goals --mastery of the skills whose practice makes up the bulk of 
a traditional secondary school mathematics program--is, of course, a 
sine qua non for further progress in mathenaatics. Efforts expended toward 
the second goal have turned out, perhaps rather unexpectedly, to provide 
strong motivation for attaining the first. UICSM students of all degrees 
of mathematical aptitude have fovmd the discovery and proof of theorems 
to be an exciting and rew^arding experience. But, they also find that errors 
in manipulation place barriers in the way of discovering provable theorenns! 

Work toward either of these goals thus helps in attaining the other, 
and the attainment of either is easier for one who has a clear understanding 
of some of the purposes for which letters are used in mathematical language. 
Two of these are particularly relevant to the work of Unit 2: the use of 
letters as real (or: free) variables, and their use as apparent (or: bound) 
variables. For example, in the expression: 

c 

(t + 2y)^dt 



•20- 






•if!.! ■>■. "..■ :.■■■: I ■ . •■■ • 

■ '•' ■•;;•■ .;;:?7'.M' >'!.; i^- •■ :: ■ 

f'i y ii.'-i/.iy.o!.;-.;. 



4>-. i 



•I u. 



- -H 



'!!': -""U ^.--i.- > 



^; ; ' I , 



^.. 13<; 



.Ml -•■1, J 



■ ••.•i-'V.-.-i'yyj ■■>i.-l: 

,--r;'';Y. 5':.? , 



, . -l-. 



(iTt'ii,-..: 



the 'x' and the 'y' are free ("really variables") and the 't's are bound ('*only 
apparent ly variables"). The 'x' naarks a position that is open to substitution; 
the 't's do not, but can, without changing the meaning of the expression, be 
replaced by occurrences of any other "dummy" symbol. (Since the above 
use of 'real variable' conflicts with its conventional meaning in, say, 'theory 
of fvmctions of a real variable', it is not introduced into the text. However. 
it will be convenient to continue this usage in what follows . ) 



Since a real variable naerely marks a place where substitutions "can* 
be made (i.e., an argument-place) and, once its domain is specified, de- 
limits the class of expressions that can be substituted for it, it turns out to 
be enlightening to simulate the notion of free variable by using actual holes 
in the paper. For example: 




[See Unit 2, Introduction. ] , The notion of substitution is conveyed by writing 
a ntimeral on a second sheet of paper and placing this sheet under the first 
so that the numeral appears in the hole. The next step is to use frames 

( I I , /\ , n/ , , etc. ) in which numerals and other appro- 



priate expressions can be written. (Such expressions may themselves con- 
tain frames and so allow for further "substitutions". ) Finally, letters are 
introduced as real variables. The first expressions using letters as vari- 
ables are those on pages 2-19 through 2-23. Parts jD and ^ help students to 
develop an understanding of algebraic "form". 

An alternative concept often introduced in preference to the concept 
of a real variable is that of an "unknown". In line with this latter concept 
the 'x' in, say, 'x + 3 = 5' is a numeral for a "definite, but unspecified" 
number. One advantage claimed for this concept is that it leads students 
to inanipulate variables in the same way as they have learned to manipulate 
(other) nunnerals. This is, of course, a desirable end, and it should be 
obvious that the sanne result is attained when such occurrences of 'x' are 
construed as real variables. Clearly, a synabol for which numerals can 
be substituted fares, during manipulation, exactly as do numerals. While 
one is symbol-pushing he treats a real variable as^_if it were a numeral. 
The inadequacy of the concept of 'x' as an unknown is first apparent when 
one considers equations such as 'x + 1 = x' and *x + x - 2 = 0'. The 
"definite but unspecified" number which 'x' is supposed to represent fails, 
in the first case, to exist, and, in the second, to be unique. Students have 
some difficulty in extending their concept of number to include such queer 
entities. The concept of 'x' as an indeterminate is, of course, something 
entirely different. It seems unlikely that this concept, valuable as it is in 
modern algebra, would be helpful to students at this stage. A fourth con- 
cept, that of a random variable, i. e, , a measurable function, is also beyond 
consideration at this level. 

Since the word 'variable' has inappropriate connotations [see Note 1, 
below], it seems desirable to use at first a word nnore descriptive of the 
actual function of real variables. Now, it is a fact that real variables serve 



■21- 



■ft ^- ■■'■'' .-- 






■•j>"ii:;i ; ■ i': 






••:'-.^v ■=•; ■'. -I ■■■■A ' 



: r, ,■ 



"ifc:.* >-to.: 



4,:. '.: l;i J ... t: 
Mil i.>'- 
lis "x" 



) i if 



./.: ty.- 



,1 ::. 'tP^ 
•If-' u 



j>lf 



one of the purposes for which pronoiins are used: they hold places for nouns. 
[Note 2] Since the free variables used in Unit 2 have domains consisting of 
numbers, and since nouns for numbers are numerals, it is helpful, at this 
stage, to coin the word 'pronumeral' to denote numerical real variables. 
When, in later units, variables with non-numerical domains make their 
appearance, the concept of real variable is sufficiently well established 
that the word 'variable' can be introduced with no danger of misunderstand- 
ing. 

The role that apparent variables play in mathematical language is also 
one played in other languages by pronouns- -and, nnore frequently, by common 
nouns. (So, in Unit 2, apparent variables are also included annong the pro- 
numerals. ) This role is that of linking operators with argunnent places. For 
example, the role of the 't's in: 

X 

(t + 2v)^dt 

is to link the operator I , d. . . with the argument- place occupied by 'u' 

•^ 1 
in the expression '(u + 2v) . [Note 3] Natural languages are quite irregular 
in the operations they use and the ways in which these are linked with argu- 
ment-places. However, the following somewhat archaic-sounding sentence 
illustrates how the operator (more specifically: the quantifier) 'each' is 
linked by means of a common noun and two pronouns to two argument-places: 

Each man, as he is honorable, so shall he prosper. 

In the same vein, the following sentence illustrates the linking of two quanti- 
fiers, each to its appropriate argument-places: 

Each man, and each woman, as he cherishes her, 
so shall she cleave to him. 

In nnathematical language constructions similar to those in the two 
examples above are used in the statements of two of the basic principles 
for real numbers: 

For each x, x + = x 

and: 

For each x, for each y, xy = yx. 

About half-way through the text of Unit 2 (but usually earlier in the class- 
room) these principles are rewritten as: 

V X + = X and: V V xy = yx. 
X X y 

The discussion and exercises leading up to such use of apparent variables 
occur on pages 2-23 through 2-27 of Unit 2. [Note 4] 

Having learned linguistic devices which make it possible to state their 
discoveries about real nunnbers concisely it is natural for students to wonder 
how nnany such discoveries they need nnake. Is it possible that some can be 
justified, or even predicted, on the basis of others? It is easy to see that 
one can disprove a generalization by finding a counter-example; but, how can 



•22- 



■liii! U' vi- 
.<■-'■■ 



or;.., ' 






■ ^ S is .-S-.V-i f" >■■ 



'j.-' VC iii.JCtUCVA.* 



•1; ». 






^ii^ f-5 .,.• 



<?**■•■ ^ 



"K^i 



•-'c,' 



.?-> (• 



• i- \ lii^K-, 



■i>is. 



V ,.) 



one prove a generalization? If one equates the property of being provable 
with that of following logically from the basic principles, it turns out that 
a generalization is provable if one has a uniform method for showing that 
each of its instances follows from appropriate instances of the basic prin- 
ciples. [Pages 2-31, 32, 33; TC[2-31, 32]a, b, c, and d] Such a method 
for showing that instances of, for exanaple: 

V 3(x2) = 6x 

X 

follow from basic principles is illustrated in the case of the instance 
'3(5* 2) = 6*5' by: 

3(5*2) = 3(2 • 5) [connmutative principle for multiplication] 

3(2*5) = (3*2)5 [associative principle for multiplication] 

(3*2)5 =6*5 [3*2=6] 

Hence, 3(5*2) =6* 5. 

(In Unit 2, the fact that 3*2 =6 is accepted as a "computing fact"; the deri- 
vation of such computing facts from definitions, such as '2 = 1 + 1', and the 
basic principles is taken up in Unit 7. ) 

Using, say, 'a' as a real variable, the testing method illustrated 
above can be indicated by a test -pattern: 

3(a'2) = 3(2* a) [cpm] 

3(2 "a) = (3* 2)a [apm] 

(3* 2)a = 6a [3*2=6] 

Hence, 3(a«2) = 6a. 

The foregoing is conceived as a pattern which can be used to test any instance 
of 'V^ 3(x2) = 6x', For example, all that is needed to show that, say, 
'3(7* 2) = 6 * 7'is a consequence of appropriate instances of the basic prin- 
ciples is to substitute '7' for 'a'. The test-pattern can be used to confound 
anyone who claims to have a counter-example to the generalization and, so, 
merits being adnnitted as a proof of the generalization. [Note 5] 

If one compares the foregoing test-pattern with the "work" expected 
of a beginning student who is to "simplify" '3(x* 2)' to *6x' the connection 
between simplifying algebraic expressions and proving elementary theorenns 
about fields becomes evident. Of more imnnediate import is the fact that 
students can learn to use their knowledge of basic principles both to discover 
computational shortcuts and to discover errors in procedures (such as sim- 
plifying '1 + 2x' to '3x') which they have adopted in hope, but which have 
proved to lead to disaster. 

In connection with simplifying expressions, the notion of equivalent 
expressions is introduced. [Pages 2-49, 50, 51; TC[2-5l]a] 

After considerable practice in sinnplifying algebraic expressions-- 
both through strict adherence to the basic principles and by the free-wheeling 
methods which successful application of the fornner procedure suggest--stu- 
dents are brought back to something nearer mathematics than symbol- pushing 



-23- 



I*.- 



■ :) ■•" •:■■ 



'■. -KIT .l'>\ !.l'? 



:„:. ,|r:i: .soLr 






in:.' > ''• 



by a short discussion on theorems and basic principles and an opportunity 
to organize their knowledge of subtraction and division. [Pages 2-60, 61, 
and TC] In the remainder of Unit 2, students discover and prove a con- 
siderable number of theorems. [These are collected in the COMMENTARY 
for page 2-6 1.] Examples: 

V x» = 

X 

VV ifx + y = then -x = y 

X y ' ' 

VV -(x-y)=y-x 

X y ' ' 

V V V x(y - z) = xy - xz 
X y z -^ ' 

V V if xy = then x = or y = 
X y ■' 

VV/VV/ ^+^ = xvJ_uiL 
xy/OuvfOy v yv 

V V / -- = — ^ 

X y ;^ y y 

Interspersed with this "theoretical" development are numerous simplification 
exercises on which students can perfect the symbol-pushing techniques whose 
justification lies in the theorems they have proved. 

'f 

Summarizing Units 1 and 2, it should be noted that students begin by 
becoming acquainted with the real numbers through conmparing and contrast- 
ing the ways in which they and the mcLre fanniliar numbers of arithmetic can 
be used in solving physical problems. On the basis of this acquaintance 
they accept sonne basic principles which describe the basic properties of 
the fundamental operations on real numbers. Having learned to use variables 
and quantifiers in order to give concise statements of these principles, they 
next learn how to derive theorems from them. Along the way, the proce- 
dures used in leading students to discover nnuch of this theoretical structure 
for themselves give ample opportunity for them to develop the manipulating 
skills whose rationale is this basic structure and which are needed in fur- 
thering its developmient. At all times students feel that they are discussing 
a "real" subject nnatter--the real nunnbers--but their actual procedure is 
much like that of one who would abstract the notion of a field from examples 
like that furnished by the real number system, and, having done so, would 
proceed to develop the elementary theory of this kind of mathematical 
structure. This early introduction of proof pays off throughout the course, 
and particularly in the deductive developnnent of Euclidean plane geometry 
in Unit 6. 



Notes 

1. A real variable is a symbol and is no more subject to change 
than is any other physical object. Hence the word 'variable' tends to be 



■24- 



"■' '.}*[.'■ ■■ ■■■ 



!.•-./ 






(■ xj„; 






ij j^ii 



■1',.". O'l '*'' 



V V 



;i : I'H.'i " 



»::»•• U J 



! ;.-•• • • . ; ■ I"!':, i •'• 'Ijf"! 

•I 

; ; Oil ^.5i--i'i " -v J' ''"<-> 

.-•..'-. 3n.">*'. .r-.'iftV'; *?■ 






misleading. On this subject I can't refrain from quoting from Professor 
E. J. McShane's Theory of Limits (MAA Film Manual, No. 2, p. 3). Dis- 
cussing common misconceptions concerning the limit concept, Professor 
McShane writes: 

Incidentally, all these offside notions have one bad 
feature in common. They all involve the idea that b; 
is "doing sonnething". . . . There is something allur- 
ing about the idea that j has a personality and "goes 
from 1 to 2 to 3" and so on, and that b- "does" some- 
thing like getting closer to 0. But this trick of per- 
sonifying j and b^ is misleading even for sequences, 
and in more complex situations it is worse. 

In case you have been thinking of the b- as a brisk 
little thing, jumping from b^ to h^ to b^, and so on, 
don't blush too hard. Not so long ago that was a pretty 
customary way of thinking of it, and the custom died 
hard. 

2. Here is an anausing exannple of a use of pronouns that strictly 
parallels the use of free variables in equation- solving problems: 

Identify the person described by the following sentences: 

He was a president of the United States. 

He conamanded American arnned forces. 

He has (had) a name consisting of six letters. 

He has (had) another name consisting of ten letters. 

He died in the nineteenth century. 

The word 'he' is, here, a real variable w^hose domain is the set of all male 
human beings (living or not). The problem is to find such a person who 
satisfies (is a solution of) all five sentences. 

3. This linking function of variables is well expounded by Quine in 
his Mathematical Logic (Cambridge, 1958), pp. 67-71. As Quine points 
out, the concept goes back at least to Peano's Formulaire of 1897, ajid was 
also exploited by Moses Schonfinkel in his "Uber die Bausteine der Mathe- 
matischen Logik" ( Math . Annalen , vol. 92 (1924), pp. 305-316). Schon- 
finkel' s position is that 

. . . the variable in a logical proposition serves 
only as a mark distinctive of certain argument- 
places and operators as mutually relevant. . . . 

4. The inadequacy of using real variables (instead of quantifiers 
and apparent variables) in stating generalizations should be apparent to 
anyone who has attempted to teach the distinction between, say, continuity 
at each point of a set and uniform continuity on that set. If the use of quan- 
tifiers and apparent variables turned out to be difficult for students to 
master, one might argue for reserving these concepts for a course in 
function theory. However, a great deal of experience shows that UICSM 
students have no particular difficulty with these concepts. Early famili- 
arity with them should place those students who continue in mathematics 



■25- 



.". ■/,:.,.■ 



:); •.• • 



v.? i, 






in a good position to appreciate the pointwise -uniform distinction. Actually, 
UICSM is more interested in the great majority of students who will not 
develop into naathematicians. Attention to linguistic nnatters such as the 
various roles played by variables appears not only to make it easier for 
such students to make sense out of mathennatics, but also sharpens their 
appreciation for correct use of language, generally. 

5. It should perhaps be noted that, while '3(5*2) = 3(2' 5)' is not 
an instance of the cpm, it is a consequence of the instance '5 • 2 = 2*5'. 
This is the case because ' = ' refers to the logical relation of identity and, 
so, multiplication having been accepted as an operation, 'V V V if x = y 
then zx = zy' is a tautology. - -H. E. V. ^ 



NEWS AND NOTICES 

The following publicatio"ns and reprints of articles are now available 
upon request from the UICSM project office: 

UICSM Information Sheet . 

Max Beberman. "Improving High School Mathematics Teaching. " 
Reprinted from Educational Leadership , December, 1959. 

William Hale. "UICSM's Decade of Experimentation." 
Preprinted from The Mathematics Teacher . 

Gertrude Hendrix. "Variable Paradox- -A Dialog in One Act. " 

Reprinted from School Science and Mathenaatics , June, 1959. 

Gertrude Hendrix. "Learning by Discovery." 

Reprinted from The Mathematics Teacher , May, 1961. 

M. Eleanor McCoy. "A Secondary School Mathamatics Progrann. " 
Reprinted from The Bulletin of the National Association of 
Secondary School Principals , May, 1959. 

UICSM Staff. "Words, 'Words', "Words"." 

Reprinted fromi The Mathennatics Teacher , March, 1957. 

UICSM Staff. "Arithmetic With Frames." 

Reprinted fronn The Mathematics Teacher , April, 195 7. 

Table of Contents, Units 1-8, HIGH SCHOOL IvIATHEMATICS. 



26- 



,Vli^' 



•i*' 



r.q-:s 



.•J- 



. v". 



.(^ 



'4* 



Correction : There is an error on page TC[7-134]d of the commentary for 
Unit 7. Line 4 should begin: 

and only if ^ is associative and has 



Mr, Ralph Futrell of Catalina High School, Tucson, Arizona, reports 
that he and Mrs. Katherine Sasse of Pueblo High School in Tucson are con- 
ducting UICSM courses for teachers in their city. Mrs. Sasse is teaching 
First Course to 22 people, while Mr. Futrell has a group of ten studying 
Units 5 and 6. He notes that "There is a greater amount of interest in 
UICSM than ever before here in Tucson. " 

Mr. Howard Marston of the Principia School, St. Louis, Missouri, 
reviewed Unit 5, Relations and Functions, of HIGH SCHOOL MATHEMATICS 
in the October, 1961, issue 6f The Mathennatics Teacher , pages 456 and 457. 



Mr. Beberman's speeches and visits during September and October 
included the following: 

September 15, 16 SA4SG panel on tests. New York City. 



September 20 
October 9 

October 9 
October 14 



Spoke to the Tazewell County, Illinois, branch of the 
American Association of University Women, at Pekin, 

Long distance telephone speech, discussing the 
Illinois program for the district meetings of the 
Idaho Education Association. 

Chicago Elementary Teachers Club, Chicago. 

Panelist at annual meeting of the Illinois Council of 
Teachers of Mathematics, Urbana. 



October 16, 17, 18 Osgood Hill Conference, Boston University, 

Andover, Mass. 



October 19 
October 20 



Visited Matignon High School, Cannbridge, Mass. 

Central Western Zone of the New York State 
Teachers Association. 



October 27 



Northwestern Ohio Teachers Association, Toledo. 



•27- 



: ■ I . .r, tft- 



co-rc'-' j.r.i'-s.: 






&/h if' ri- ■•;-;.., . , : n't.- ^'5 

.n.b;-.''^^ l!i .loir? '-J- 



•Ji'. i J^fU ^ 



•■ t; V ^ • 



-.A rt- 



Now Available 

TESTS 

FOR UNITS 1 THROUGH 4 

OF HIGH SCHOOL MATHEMATICS 

Prepared by the University of Illinois Committee on School Mathematics 

Pupil examinations for Units 1 through 4 of High School Mathematics , pre- 
pared by the University of Illinois Committee on School Mathematics, are 
now available from the University of Illinois Press. 

A total of five examinations has been designed to test pupils in classes using 
the official texts for Units 1 through 4 of High School Mathematics . There 
is an examination for each of these first four units, plus a comprehensive 
final exanriination covering all four units. 

Each examination is a four-page folder containing a 24-item, multiple-choice 
test designed for 30-minute testing time. There is only one correct choice 
for each item. The University of Illinois Committee on School Mathematics 
suggests that teachers have' pupils mark their answers to a test on a separate 
answer sheet and that a uniform answer sheet, listing the numbers 1 through 
24, be prepared to simplify scoring. In this way, the exannination folders 
may be reused indefinitely. For mass testing, schools nnay want to purchase 
24-item {5-choice) answer cards from the International Business Machines 
Corporation. 

Each of the five tests for Units 1 through 4 are packaged 35 copies to a mail- 
ing envelope. Each envelope also contains a four-page folder giving exami- 
nation instructions, keys, and norms for eighth and ninth grades. 

These tests will be sold only in nnultiples of 35 copies . Orders listing quan - 
tities not in multiples of 35 copies will be filled according to the closest 
mviltiple of 35 . 

The price of each package of 35 tests is $1. 75. There is a discount of 10 
per cent if all five tests for Units 1 through 4 are ordered together in the 
same quantities. 

Please use the following descriptions when ordering tests: 
Quantity Test 

Unit 1 Examination, UICSM - High School Mathematics 

Unit 2 Examination, UICSM - High School Mathematics 

Unit 3 Examination, UICSM - High School Mathematics 

Unit 4 Examination, UICSM - High School Mathematics 

Final Examination (Units 1-4), UICSM - High School Mathematics 

All orders must be on school or board of education purchase orders or 
letterheads . Orders received in any other way will not be accepted . 

Send all orders directly to: University of Illinois Press, Urbana, Illinois. 

Examinations for Unit 5, Unit 6, and future units will be announced as they 
become available. 

10/3/60 --The University of Illinois Press 

-28- 



;J •' 



•J. ' 'll' 









NOTICE 






If 


you wish to continue receiving this newslett 


er, 


complete 


the form 


below and return this 


page 


to: 






Editor, 


UICSM Newsletter 










1208 We 


St Springfield 










Urbana, 


Illinois 







Please send me the UICSM Newsletter during the academic year 1961-62, 



Name 



Address 



Please note below when and where you were introduced to or studied the 
UICSM program, whether in an NSF Atademic Year Institute, Summer 
Institute, in-service training, methods, curriculum, or extension course, 
etc. 



As suggested on Page 2, please list the nannes and addresses of others 
whom you feel should receive copies of the Newsletter and are qualified 
to read it with \inderstanding. 



-29- 



litr- ; Vr-.' i 



■^i-^ajv H;f:r^^-:^ =)ri.fc ;y.:.7«'^ if:--' '